Theorem poimirlem9 38011

Description: Lemma for poimir 38035, establishing the two walks that yield a given face when the opposite vertex is neither first nor last. (Contributed by Brendan Leahy, 21-Aug-2020.)

Hypotheses

Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem22.s	⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem9.1	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem9.2	⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))
poimirlem9.3	⊢ (𝜑 → 𝑈 ∈ 𝑆)
poimirlem9.4	⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) ≠ (2nd ‘(1st ‘𝑇)))

Assertion

Ref	Expression
poimirlem9	⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) = ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))))

Distinct variable groups:   𝑓,𝑗,𝑡,𝑦   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑁,𝑦   𝑇,𝑗,𝑦   𝑈,𝑗,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑁,𝑡   𝑇,𝑓   𝑈,𝑓   𝑓,𝐹,𝑡   𝑡,𝑇   𝑡,𝑈   𝑆,𝑗,𝑡,𝑦

Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem9

Step	Hyp	Ref	Expression
1		resundi 5952	. . . 4 ⊢ ((2nd ‘(1st ‘𝑈)) ↾ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
2		poimir.0	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	nncnd 12185	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
4		npcan1 11570	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5	3, 4	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
6	2	nnzd 12545	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		peano2zm 12565	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
8		uzid 12798	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
9		peano2uz 12846	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
10	6, 7, 8, 9	4syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
11	5, 10	eqeltrrd 2842	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
12		fzss2 13513	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
14		poimirlem9.2	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))
15	13, 14	sseldd 3918	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...𝑁))
16		fzp1elp1 13526	. . . . . . . . . 10 ⊢ ((2nd ‘𝑇) ∈ (1...(𝑁 − 1)) → ((2nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
18	5	oveq2d 7376	. . . . . . . . 9 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
19	17, 18	eleqtrd 2843	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) ∈ (1...𝑁))
20	15, 19	prssd 4756	. . . . . . 7 ⊢ (𝜑 → {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ⊆ (1...𝑁))
21		undif 4413	. . . . . . 7 ⊢ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ⊆ (1...𝑁) ↔ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = (1...𝑁))
22	20, 21	sylib 220	. . . . . 6 ⊢ (𝜑 → ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = (1...𝑁))
23	22	reseq2d 5938	. . . . 5 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = ((2nd ‘(1st ‘𝑈)) ↾ (1...𝑁)))
24		poimirlem9.3	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
25		elrabi 3627	. . . . . . . . 9 ⊢ (𝑈 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
26		poimirlem22.s	. . . . . . . . 9 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
27	25, 26	eleq2s 2859	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
28		xp1st 7967	. . . . . . . 8 ⊢ (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
29		xp2nd 7968	. . . . . . . 8 ⊢ ((1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
30	24, 27, 28, 29	4syl 19	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
31		fvex 6844	. . . . . . . 8 ⊢ (2nd ‘(1st ‘𝑈)) ∈ V
32		f1oeq1 6759	. . . . . . . 8 ⊢ (𝑓 = (2nd ‘(1st ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
33	31, 32	elab 3619	. . . . . . 7 ⊢ ((2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
34	30, 33	sylib 220	. . . . . 6 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
35		f1ofn 6772	. . . . . 6 ⊢ ((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑈)) Fn (1...𝑁))
36		fnresdm 6608	. . . . . 6 ⊢ ((2nd ‘(1st ‘𝑈)) Fn (1...𝑁) → ((2nd ‘(1st ‘𝑈)) ↾ (1...𝑁)) = (2nd ‘(1st ‘𝑈)))
37	34, 35, 36	3syl 18	. . . . 5 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ (1...𝑁)) = (2nd ‘(1st ‘𝑈)))
38	23, 37	eqtrd 2776	. . . 4 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (2nd ‘(1st ‘𝑈)))
39	1, 38	eqtr3id 2790	. . 3 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (2nd ‘(1st ‘𝑈)))
40		2lt3 12343	. . . . . 6 ⊢ 2 < 3
41		2re 12250	. . . . . . 7 ⊢ 2 ∈ ℝ
42		3re 12256	. . . . . . 7 ⊢ 3 ∈ ℝ
43	41, 42	ltnlei 11262	. . . . . 6 ⊢ (2 < 3 ↔ ¬ 3 ≤ 2)
44	40, 43	mpbi 232	. . . . 5 ⊢ ¬ 3 ≤ 2
45		df-pr 4561	. . . . . . . . . . . 12 ⊢ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} = ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩} ∪ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})
46	45	coeq2i 5805	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) = ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩} ∪ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}))
47		coundi 6202	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩} ∪ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) = (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}) ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}))
48	46, 47	eqtri 2764	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) = (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}) ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}))
49		poimirlem9.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
50		elrabi 3627	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
51	50, 26	eleq2s 2859	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
52		xp1st 7967	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
53		xp2nd 7968	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
54	49, 51, 52, 53	4syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
55		fvex 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
56		f1oeq1 6759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
57	55, 56	elab 3619	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
58	54, 57	sylib 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
59		f1of1 6770	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁))
60	58, 59	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁))
61	19	snssd 4721	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {((2nd ‘𝑇) + 1)} ⊆ (1...𝑁))
62		f1ores 6785	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ {((2nd ‘𝑇) + 1)} ⊆ (1...𝑁)) → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}–1-1-onto→((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)}))
63	60, 61, 62	syl2anc 591	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}–1-1-onto→((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)}))
64		f1of 6771	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}–1-1-onto→((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)}) → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}⟶((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)}))
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}⟶((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)}))
66		f1ofn 6772	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
67	58, 66	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) Fn (1...𝑁))
68		fnsnfv 6910	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((2nd ‘𝑇) + 1) ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} = ((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)}))
69	67, 19, 68	syl2anc 591	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} = ((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)}))
70	69	feq3d 6644	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} ↔ ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}⟶((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1)})))
71	65, 70	mpbird 259	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))})
72		eqid 2741	. . . . . . . . . . . . . . 15 ⊢ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩} = {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}
73		fvex 6844	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝑇) ∈ V
74		ovex 7393	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑇) + 1) ∈ V
75	73, 74	fsn 7081	. . . . . . . . . . . . . . 15 ⊢ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}:{(2nd ‘𝑇)}⟶{((2nd ‘𝑇) + 1)} ↔ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩} = {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩})
76	72, 75	mpbir 233	. . . . . . . . . . . . . 14 ⊢ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}:{(2nd ‘𝑇)}⟶{((2nd ‘𝑇) + 1)}
77		fco2 6685	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}):{((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} ∧ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}:{(2nd ‘𝑇)}⟶{((2nd ‘𝑇) + 1)}) → ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}):{(2nd ‘𝑇)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))})
78	71, 76, 77	sylancl 593	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}):{(2nd ‘𝑇)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))})
79		fvex 6844	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) ∈ V
80	79	fconst2 7153	. . . . . . . . . . . . 13 ⊢ (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}):{(2nd ‘𝑇)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} ↔ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}) = ({(2nd ‘𝑇)} × {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))}))
81	78, 80	sylib 220	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}) = ({(2nd ‘𝑇)} × {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))}))
82	73, 79	xpsn 7087	. . . . . . . . . . . 12 ⊢ ({(2nd ‘𝑇)} × {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))}) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}
83	81, 82	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
84	83	uneq1d 4100	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩}) ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) = ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})))
85	48, 84	eqtrid 2788	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) = ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})))
86		elfznn 13502	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑇) ∈ ℕ)
87	14, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2nd ‘𝑇) ∈ ℕ)
88	87	nnred 12184	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘𝑇) ∈ ℝ)
89	88	ltp1d 12081	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘𝑇) < ((2nd ‘𝑇) + 1))
90	88, 89	ltned 11277	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2nd ‘𝑇) ≠ ((2nd ‘𝑇) + 1))
91	90	necomd 2991	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2nd ‘𝑇) + 1) ≠ (2nd ‘𝑇))
92		f1veqaeq 7204	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ (((2nd ‘𝑇) + 1) ∈ (1...𝑁) ∧ (2nd ‘𝑇) ∈ (1...𝑁))) → (((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) = ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) → ((2nd ‘𝑇) + 1) = (2nd ‘𝑇)))
93	60, 19, 15, 92	syl12anc 843	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) = ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) → ((2nd ‘𝑇) + 1) = (2nd ‘𝑇)))
94	93	necon3d 2957	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2nd ‘𝑇) + 1) ≠ (2nd ‘𝑇) → ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) ≠ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))))
95	91, 94	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) ≠ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)))
96	95	neneqd 2941	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) = ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)))
97	73, 79	opth 5419	. . . . . . . . . . . 12 ⊢ (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ ↔ ((2nd ‘𝑇) = (2nd ‘𝑇) ∧ ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) = ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))))
98	97	simprbi 499	. . . . . . . . . . 11 ⊢ (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ → ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) = ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)))
99	96, 98	nsyl 140	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩)
100	90	neneqd 2941	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (2nd ‘𝑇) = ((2nd ‘𝑇) + 1))
101	73, 79	opth1 5418	. . . . . . . . . . 11 ⊢ (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ → (2nd ‘𝑇) = ((2nd ‘𝑇) + 1))
102	100, 101	nsyl 140	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩)
103		opex 5406	. . . . . . . . . . . . . . 15 ⊢ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ V
104	103	snid 4597	. . . . . . . . . . . . . 14 ⊢ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}
105		elun1 4114	. . . . . . . . . . . . . 14 ⊢ (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} → ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})))
106	104, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}))
107		eleq2 2830	. . . . . . . . . . . . 13 ⊢ (({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} → (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) ↔ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}))
108	106, 107	mpbii 235	. . . . . . . . . . . 12 ⊢ (({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} → ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
109	103	elpr 4583	. . . . . . . . . . . . 13 ⊢ (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ↔ (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ ∨ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩))
110		oran 998	. . . . . . . . . . . . 13 ⊢ ((⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ ∨ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩) ↔ ¬ (¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ ∧ ¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩))
111	109, 110	bitri 277	. . . . . . . . . . . 12 ⊢ (⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ ∈ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ↔ ¬ (¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ ∧ ¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩))
112	108, 111	sylib 220	. . . . . . . . . . 11 ⊢ (({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} → ¬ (¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ ∧ ¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩))
113	112	necon2ai 2965	. . . . . . . . . 10 ⊢ ((¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩ ∧ ¬ ⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩ = ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩) → ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) ≠ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
114	99, 102, 113	syl2anc 591	. . . . . . . . 9 ⊢ (𝜑 → ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∪ ((2nd ‘(1st ‘𝑇)) ∘ {⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) ≠ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
115	85, 114	eqnetrd 3003	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ≠ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
116		fnressn 7105	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (2nd ‘𝑇) ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇)}) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩})
117	67, 15, 116	syl2anc 591	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇)}) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩})
118		fnressn 7105	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((2nd ‘𝑇) + 1) ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}) = {⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
119	67, 19, 118	syl2anc 591	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}) = {⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
120	117, 119	uneq12d 4102	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇)}) ∪ ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)})) = ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩} ∪ {⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}))
121		df-pr 4561	. . . . . . . . . . . 12 ⊢ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} = ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)})
122	121	reseq2i 5935	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((2nd ‘(1st ‘𝑇)) ↾ ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)}))
123		resundi 5952	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)) ↾ ({(2nd ‘𝑇)} ∪ {((2nd ‘𝑇) + 1)})) = (((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇)}) ∪ ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}))
124	122, 123	eqtri 2764	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = (((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇)}) ∪ ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1)}))
125		df-pr 4561	. . . . . . . . . 10 ⊢ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} = ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩} ∪ {⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
126	120, 124, 125	3eqtr4g 2801	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩})
127		poimirlem9.4	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) ≠ (2nd ‘(1st ‘𝑇)))
128	2, 26, 49, 14, 24	poimirlem8 38010	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
129		uneq12 4096	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∧ ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
130		resundi 5952	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑇)) ↾ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
131	22	reseq2d 5938	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = ((2nd ‘(1st ‘𝑇)) ↾ (1...𝑁)))
132		fnresdm 6608	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) → ((2nd ‘(1st ‘𝑇)) ↾ (1...𝑁)) = (2nd ‘(1st ‘𝑇)))
133	58, 66, 132	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ (1...𝑁)) = (2nd ‘(1st ‘𝑇)))
134	131, 133	eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (2nd ‘(1st ‘𝑇)))
135	130, 134	eqtr3id 2790	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (2nd ‘(1st ‘𝑇)))
136	39, 135	eqeq12d 2757	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) ↔ (2nd ‘(1st ‘𝑈)) = (2nd ‘(1st ‘𝑇))))
137	129, 136	imbitrid 246	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∧ ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) → (2nd ‘(1st ‘𝑈)) = (2nd ‘(1st ‘𝑇))))
138	128, 137	mpan2d 701	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) → (2nd ‘(1st ‘𝑈)) = (2nd ‘(1st ‘𝑇))))
139	138	necon3d 2957	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ≠ (2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
140	127, 139	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
141	140	necomd 2991	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
142	126, 141	eqnetrrd 3004	. . . . . . . 8 ⊢ (𝜑 → {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ≠ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
143		prex 5370	. . . . . . . . . . . 12 ⊢ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∈ V
144	55, 143	coex 7874	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ∈ V
145		prex 5370	. . . . . . . . . . 11 ⊢ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∈ V
146	31	resex 5988	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∈ V
147		hashtpg 14442	. . . . . . . . . . 11 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ∈ V ∧ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∈ V ∧ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∈ V) → ((((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ≠ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∧ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ≠ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∧ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) ↔ (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) = 3))
148	144, 145, 146, 147	mp3an 1470	. . . . . . . . . 10 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ≠ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∧ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ≠ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∧ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) ↔ (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) = 3)
149	148	biimpi 218	. . . . . . . . 9 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ≠ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∧ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ≠ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∧ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})) → (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) = 3)
150	149	3expia 1128	. . . . . . . 8 ⊢ ((((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ≠ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ∧ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} ≠ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) → (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) = 3))
151	115, 142, 150	syl2anc 591	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) → (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) = 3))
152		prex 5370	. . . . . . . . . . . . 13 ⊢ {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∈ V
153		prex 5370	. . . . . . . . . . . . 13 ⊢ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∈ V
154	152, 153	mapval 8779	. . . . . . . . . . . 12 ⊢ ({((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↑m {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}
155		prfi 9228	. . . . . . . . . . . . 13 ⊢ {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∈ Fin
156		prfi 9228	. . . . . . . . . . . . 13 ⊢ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∈ Fin
157		mapfi 9252	. . . . . . . . . . . . 13 ⊢ (({((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∈ Fin ∧ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∈ Fin) → ({((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↑m {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∈ Fin)
158	155, 156, 157	mp2an 699	. . . . . . . . . . . 12 ⊢ ({((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↑m {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∈ Fin
159	154, 158	eqeltrri 2838	. . . . . . . . . . 11 ⊢ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ∈ Fin
160		f1of 6771	. . . . . . . . . . . 12 ⊢ (𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} → 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
161	160	ss2abi 4000	. . . . . . . . . . 11 ⊢ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}
162		ssfi 9101	. . . . . . . . . . 11 ⊢ (({𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ∈ Fin ∧ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}) → {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ∈ Fin)
163	159, 161, 162	mp2an 699	. . . . . . . . . 10 ⊢ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ∈ Fin
164	19, 15	prssd 4756	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {((2nd ‘𝑇) + 1), (2nd ‘𝑇)} ⊆ (1...𝑁))
165		f1ores 6785	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)} ⊆ (1...𝑁)) → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}):{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}–1-1-onto→((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}))
166	60, 164, 165	syl2anc 591	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}):{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}–1-1-onto→((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}))
167		fnimapr 6914	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((2nd ‘𝑇) + 1) ∈ (1...𝑁) ∧ (2nd ‘𝑇) ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) = {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
168	67, 19, 15, 167	syl3anc 1380	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) = {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
169	168	f1oeq3d 6768	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}):{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}–1-1-onto→((2nd ‘(1st ‘𝑇)) “ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) ↔ ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}):{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
170	166, 169	mpbid 234	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}):{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
171		f1oprg 6817	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝑇) ∈ V ∧ ((2nd ‘𝑇) + 1) ∈ V) ∧ (((2nd ‘𝑇) + 1) ∈ V ∧ (2nd ‘𝑇) ∈ V)) → (((2nd ‘𝑇) ≠ ((2nd ‘𝑇) + 1) ∧ ((2nd ‘𝑇) + 1) ≠ (2nd ‘𝑇)) → {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}))
172	73, 74, 74, 73, 171	mp4an 700	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑇) ≠ ((2nd ‘𝑇) + 1) ∧ ((2nd ‘𝑇) + 1) ≠ (2nd ‘𝑇)) → {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘𝑇) + 1), (2nd ‘𝑇)})
173	90, 91, 172	syl2anc 591	. . . . . . . . . . . . 13 ⊢ (𝜑 → {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘𝑇) + 1), (2nd ‘𝑇)})
174		f1oco 6794	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}):{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∧ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) → (((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
175	170, 173, 174	syl2anc 591	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
176		rnpropg 6177	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑇) ∈ V ∧ ((2nd ‘𝑇) + 1) ∈ V) → ran {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} = {((2nd ‘𝑇) + 1), (2nd ‘𝑇)})
177	73, 74, 176	mp2an 699	. . . . . . . . . . . . . 14 ⊢ ran {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} = {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}
178	177	eqimssi 3977	. . . . . . . . . . . . 13 ⊢ ran {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ⊆ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}
179		cores 6204	. . . . . . . . . . . . 13 ⊢ (ran {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ⊆ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)} → (((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) = ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}))
180		f1oeq1 6759	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) = ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) → ((((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↔ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
181	178, 179, 180	mp2b 10	. . . . . . . . . . . 12 ⊢ ((((2nd ‘(1st ‘𝑇)) ↾ {((2nd ‘𝑇) + 1), (2nd ‘𝑇)}) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↔ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
182	175, 181	sylib 220	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
183	95	necomd 2991	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ≠ ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)))
184		fvex 6844	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ∈ V
185		f1oprg 6817	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝑇) ∈ V ∧ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ∈ V) ∧ (((2nd ‘𝑇) + 1) ∈ V ∧ ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) ∈ V)) → (((2nd ‘𝑇) ≠ ((2nd ‘𝑇) + 1) ∧ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ≠ ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))) → {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))}))
186	73, 184, 74, 79, 185	mp4an 700	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑇) ≠ ((2nd ‘𝑇) + 1) ∧ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ≠ ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))) → {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))})
187	90, 183, 186	syl2anc 591	. . . . . . . . . . . 12 ⊢ (𝜑 → {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))})
188		prcom 4667	. . . . . . . . . . . . 13 ⊢ {((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} = {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}
189		f1oeq3 6761	. . . . . . . . . . . . 13 ⊢ ({((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} = {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} → ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} ↔ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
190	188, 189	ax-mp 5	. . . . . . . . . . . 12 ⊢ ({⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))} ↔ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
191	187, 190	sylib 220	. . . . . . . . . . 11 ⊢ (𝜑 → {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
192		f1of1 6770	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1→(1...𝑁))
193	34, 192	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1→(1...𝑁))
194		f1ores 6785	. . . . . . . . . . . . 13 ⊢ (((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1→(1...𝑁) ∧ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ⊆ (1...𝑁)) → ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→((2nd ‘(1st ‘𝑈)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
195	193, 20, 194	syl2anc 591	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→((2nd ‘(1st ‘𝑈)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
196		dff1o3 6777	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st ‘𝑈))))
197	196	simprbi 499	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st ‘𝑈)))
198		imadif 6573	. . . . . . . . . . . . . . . 16 ⊢ (Fun ◡(2nd ‘(1st ‘𝑈)) → ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
199	34, 197, 198	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
200		f1ofo 6778	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁))
201		foima 6748	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑈)) “ (1...𝑁)) = (1...𝑁))
202	34, 200, 201	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) “ (1...𝑁)) = (1...𝑁))
203		f1ofo 6778	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
204		foima 6748	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
205	58, 203, 204	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
206	202, 205	eqtr4d 2779	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) “ (1...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑁)))
207	128	rneqd 5887	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ran ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
208		df-ima 5634	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ran ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
209		df-ima 5634	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ran ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
210	207, 208, 209	3eqtr4g 2801	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
211	206, 210	difeq12d 4061	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
212		dff1o3 6777	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st ‘𝑇))))
213	212	simprbi 499	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st ‘𝑇)))
214		imadif 6573	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ◡(2nd ‘(1st ‘𝑇)) → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
215	58, 213, 214	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
216		dfin4 4209	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...𝑁) ∩ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
217		sseqin2 4155	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ⊆ (1...𝑁) ↔ ((1...𝑁) ∩ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
218	20, 217	sylib 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...𝑁) ∩ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
219	216, 218	eqtr3id 2790	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
220	219	imaeq2d 6019	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = ((2nd ‘(1st ‘𝑇)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
221	215, 220	eqtr3d 2778	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = ((2nd ‘(1st ‘𝑇)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
222	199, 211, 221	3eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = ((2nd ‘(1st ‘𝑇)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
223	219	imaeq2d 6019	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = ((2nd ‘(1st ‘𝑈)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
224		fnimapr 6914	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (2nd ‘𝑇) ∈ (1...𝑁) ∧ ((2nd ‘𝑇) + 1) ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑇)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))})
225	67, 15, 19, 224	syl3anc 1380	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))})
226	225, 188	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
227	222, 223, 226	3eqtr3d 2784	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
228	227	f1oeq3d 6768	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→((2nd ‘(1st ‘𝑈)) “ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ↔ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
229	195, 228	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
230		ssabral 3998	. . . . . . . . . . . 12 ⊢ ({((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ↔ ∀𝑓 ∈ {((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))})
231		f1oeq1 6759	. . . . . . . . . . . . 13 ⊢ (𝑓 = ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) → (𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↔ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
232		f1oeq1 6759	. . . . . . . . . . . . 13 ⊢ (𝑓 = {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩} → (𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↔ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
233		f1oeq1 6759	. . . . . . . . . . . . 13 ⊢ (𝑓 = ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) → (𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↔ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
234	144, 145, 146, 231, 232, 233	raltp 4640	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ {((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ↔ (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∧ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∧ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
235	230, 234	bitri 277	. . . . . . . . . . 11 ⊢ ({((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ↔ (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∧ {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∧ ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}):{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}))
236	182, 191, 229, 235	syl3anbrc 1351	. . . . . . . . . 10 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}})
237		hashss 14366	. . . . . . . . . 10 ⊢ (({𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ∈ Fin ∧ {((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}) → (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) ≤ (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}))
238	163, 236, 237	sylancr 594	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) ≤ (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}))
239	153	enref 8926	. . . . . . . . . . . 12 ⊢ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ≈ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}
240		hashprg 14352	. . . . . . . . . . . . . . . 16 ⊢ ((((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) ∈ V ∧ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ∈ V) → (((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) ≠ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ↔ (♯‘{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}) = 2))
241	79, 184, 240	mp2an 699	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)) ≠ ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇)) ↔ (♯‘{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}) = 2)
242	95, 241	sylib 220	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}) = 2)
243		hashprg 14352	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑇) ∈ V ∧ ((2nd ‘𝑇) + 1) ∈ V) → ((2nd ‘𝑇) ≠ ((2nd ‘𝑇) + 1) ↔ (♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = 2))
244	73, 74, 243	mp2an 699	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑇) ≠ ((2nd ‘𝑇) + 1) ↔ (♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = 2)
245	90, 244	sylib 220	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = 2)
246	242, 245	eqtr4d 2779	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}) = (♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
247		hashen 14304	. . . . . . . . . . . . . 14 ⊢ (({((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ∈ Fin ∧ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∈ Fin) → ((♯‘{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}) = (♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ↔ {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ≈ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
248	155, 156, 247	mp2an 699	. . . . . . . . . . . . 13 ⊢ ((♯‘{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}) = (♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ↔ {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ≈ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
249	246, 248	sylib 220	. . . . . . . . . . . 12 ⊢ (𝜑 → {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ≈ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
250		hashfacen 14411	. . . . . . . . . . . 12 ⊢ (({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ≈ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∧ {((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))} ≈ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) → {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}})
251	239, 249, 250	sylancr 594	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}})
252	153, 153	mapval 8779	. . . . . . . . . . . . . 14 ⊢ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ↑m {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}
253		mapfi 9252	. . . . . . . . . . . . . . 15 ⊢ (({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∈ Fin ∧ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∈ Fin) → ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ↑m {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∈ Fin)
254	156, 156, 253	mp2an 699	. . . . . . . . . . . . . 14 ⊢ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ↑m {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∈ Fin
255	252, 254	eqeltrri 2838	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}} ∈ Fin
256		f1of 6771	. . . . . . . . . . . . . 14 ⊢ (𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} → 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})
257	256	ss2abi 4000	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}
258		ssfi 9101	. . . . . . . . . . . . 13 ⊢ (({𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}} ∈ Fin ∧ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}} ⊆ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}⟶{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}) → {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}} ∈ Fin)
259	255, 257, 258	mp2an 699	. . . . . . . . . . . 12 ⊢ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}} ∈ Fin
260		hashen 14304	. . . . . . . . . . . 12 ⊢ (({𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ∈ Fin ∧ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}} ∈ Fin) → ((♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}) = (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}) ↔ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}))
261	163, 259, 260	mp2an 699	. . . . . . . . . . 11 ⊢ ((♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}) = (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}) ↔ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}})
262	251, 261	sylibr 236	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}) = (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}))
263		hashfac 14415	. . . . . . . . . . . 12 ⊢ ({(2nd ‘𝑇), ((2nd ‘𝑇) + 1)} ∈ Fin → (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}) = (!‘(♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
264	156, 263	ax-mp 5	. . . . . . . . . . 11 ⊢ (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}) = (!‘(♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
265	245	fveq2d 6835	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘(♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = (!‘2))
266		fac2 14236	. . . . . . . . . . . 12 ⊢ (!‘2) = 2
267	265, 266	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘(♯‘{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = 2)
268	264, 267	eqtrid 2788	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}}) = 2)
269	262, 268	eqtrd 2776	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:{(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}–1-1-onto→{((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1)), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))}}) = 2)
270	238, 269	breqtrd 5101	. . . . . . . 8 ⊢ (𝜑 → (♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) ≤ 2)
271		breq1 5078	. . . . . . . 8 ⊢ ((♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) = 3 → ((♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) ≤ 2 ↔ 3 ≤ 2))
272	270, 271	syl5ibcom 247	. . . . . . 7 ⊢ (𝜑 → ((♯‘{((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}), {⟨(2nd ‘𝑇), ((2nd ‘(1st ‘𝑇))‘(2nd ‘𝑇))⟩, ⟨((2nd ‘𝑇) + 1), ((2nd ‘(1st ‘𝑇))‘((2nd ‘𝑇) + 1))⟩}, ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})}) = 3 → 3 ≤ 2))
273	151, 272	syld 47	. . . . . 6 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ≠ ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) → 3 ≤ 2))
274	273	necon1bd 2954	. . . . 5 ⊢ (𝜑 → (¬ 3 ≤ 2 → ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩})))
275	44, 274	mpi 20	. . . 4 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) = ((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}))
276		coires1 6220	. . . . 5 ⊢ ((2nd ‘(1st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))
277	128, 276	eqtr4di 2794	. . . 4 ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
278	275, 277	uneq12d 4102	. . 3 ⊢ (𝜑 → (((2nd ‘(1st ‘𝑈)) ↾ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}) ∪ ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))) = (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ∪ ((2nd ‘(1st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))))
279	39, 278	eqtr3d 2778	. 2 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) = (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ∪ ((2nd ‘(1st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))))
280		coundi 6202	. 2 ⊢ ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))) = (((2nd ‘(1st ‘𝑇)) ∘ {⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩}) ∪ ((2nd ‘(1st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))
281	279, 280	eqtr4di 2794	1 ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) = ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  {cab 2719   ≠ wne 2936  ∀wral 3055  {crab 3393  Vcvv 3433  ⦋csb 3833   ∖ cdif 3882   ∪ cun 3883   ∩ cin 3884   ⊆ wss 3885  ifcif 4457  {csn 4558  {cpr 4560  {ctp 4562  ⟨cop 4564   class class class wbr 5075   ↦ cmpt 5156   I cid 5515   × cxp 5619  ^◡ccnv 5620  ran crn 5622   ↾ cres 5623   “ cima 5624   ∘ ccom 5625  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485  –1-1→wf1 6486  –onto→wfo 6487  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360   ∘_f cof 7622  1^st c1st 7933  2^nd c2nd 7934   ↑_m cmap 8767   ≈ cen 8884  Fincfn 8887  ℂcc 11031  0cc0 11033  1c1 11034   + caddc 11036   < clt 11174   ≤ cle 11175   − cmin 11372  ℕcn 12169  2c2 12231  3c3 12232  ℤcz 12519  ℤ_≥cuz 12783  ...cfz 13456  ..^cfzo 13603  !cfa 14230  ♯chash 14287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-fac 14231  df-bc 14260  df-hash 14288

This theorem is referenced by:  poimirlem22  38024