Theorem erdszelem9 35184

Description: Lemma for erdsze 35187. (Contributed by Mario Carneiro, 22-Jan-2015.)

Hypotheses

Ref	Expression
erdsze.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
erdsze.f	⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i	⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.j	⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.t	⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)

Assertion

Ref	Expression
erdszelem9	⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))

Distinct variable groups:   𝑥,𝑦,𝑛,𝐹   𝑛,𝐼,𝑥,𝑦   𝑛,𝐽,𝑥,𝑦   𝑛,𝑁,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦

Allowed substitution hints:   𝑇(𝑥,𝑦,𝑛)

Proof of Theorem erdszelem9

Dummy variables 𝑤 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		erdsze.f	. . . . . 6 ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
3		erdszelem.i	. . . . . 6 ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
4		ltso 11339	. . . . . 6 ⊢ < Or ℝ
5	1, 2, 3, 4	erdszelem6 35181	. . . . 5 ⊢ (𝜑 → 𝐼:(1...𝑁)⟶ℕ)
6	5	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝐼‘𝑛) ∈ ℕ)
7		erdszelem.j	. . . . . 6 ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
8		gtso 11340	. . . . . 6 ⊢ ◡ < Or ℝ
9	1, 2, 7, 8	erdszelem6 35181	. . . . 5 ⊢ (𝜑 → 𝐽:(1...𝑁)⟶ℕ)
10	9	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝐽‘𝑛) ∈ ℕ)
11		opelxpi 5726	. . . 4 ⊢ (((𝐼‘𝑛) ∈ ℕ ∧ (𝐽‘𝑛) ∈ ℕ) → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ ∈ (ℕ × ℕ))
12	6, 10, 11	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ ∈ (ℕ × ℕ))
13		erdszelem.t	. . 3 ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)
14	12, 13	fmptd 7134	. 2 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(ℕ × ℕ))
15		fveq2 6907	. . . . . 6 ⊢ (𝑎 = 𝑧 → (𝑇‘𝑎) = (𝑇‘𝑧))
16		fveq2 6907	. . . . . 6 ⊢ (𝑏 = 𝑤 → (𝑇‘𝑏) = (𝑇‘𝑤))
17	15, 16	eqeqan12d 2749	. . . . 5 ⊢ ((𝑎 = 𝑧 ∧ 𝑏 = 𝑤) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑧) = (𝑇‘𝑤)))
18		eqeq12 2752	. . . . 5 ⊢ ((𝑎 = 𝑧 ∧ 𝑏 = 𝑤) → (𝑎 = 𝑏 ↔ 𝑧 = 𝑤))
19	17, 18	imbi12d 344	. . . 4 ⊢ ((𝑎 = 𝑧 ∧ 𝑏 = 𝑤) → (((𝑇‘𝑎) = (𝑇‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤)))
20		fveq2 6907	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑇‘𝑎) = (𝑇‘𝑤))
21		fveq2 6907	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (𝑇‘𝑏) = (𝑇‘𝑧))
22	20, 21	eqeqan12d 2749	. . . . . 6 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑤) = (𝑇‘𝑧)))
23		eqcom 2742	. . . . . 6 ⊢ ((𝑇‘𝑤) = (𝑇‘𝑧) ↔ (𝑇‘𝑧) = (𝑇‘𝑤))
24	22, 23	bitrdi 287	. . . . 5 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑧) = (𝑇‘𝑤)))
25		eqeq12 2752	. . . . . 6 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → (𝑎 = 𝑏 ↔ 𝑤 = 𝑧))
26		eqcom 2742	. . . . . 6 ⊢ (𝑤 = 𝑧 ↔ 𝑧 = 𝑤)
27	25, 26	bitrdi 287	. . . . 5 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → (𝑎 = 𝑏 ↔ 𝑧 = 𝑤))
28	24, 27	imbi12d 344	. . . 4 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → (((𝑇‘𝑎) = (𝑇‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤)))
29		elfzelz 13561	. . . . . . 7 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
30	29	zred 12720	. . . . . 6 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
31	30	ssriv 3999	. . . . 5 ⊢ (1...𝑁) ⊆ ℝ
32	31	a1i 11	. . . 4 ⊢ (𝜑 → (1...𝑁) ⊆ ℝ)
33		biidd 262	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → (((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤) ↔ ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤)))
34		simpr1 1193	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑧 ∈ (1...𝑁))
35		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝐼‘𝑛) = (𝐼‘𝑧))
36		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝐽‘𝑛) = (𝐽‘𝑧))
37	35, 36	opeq12d 4886	. . . . . . . . 9 ⊢ (𝑛 = 𝑧 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩)
38		opex 5475	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ ∈ V
39	37, 13, 38	fvmpt 7016	. . . . . . . 8 ⊢ (𝑧 ∈ (1...𝑁) → (𝑇‘𝑧) = ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩)
40	34, 39	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑇‘𝑧) = ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩)
41		simpr2 1194	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑤 ∈ (1...𝑁))
42		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑛 = 𝑤 → (𝐼‘𝑛) = (𝐼‘𝑤))
43		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑛 = 𝑤 → (𝐽‘𝑛) = (𝐽‘𝑤))
44	42, 43	opeq12d 4886	. . . . . . . . 9 ⊢ (𝑛 = 𝑤 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩)
45		opex 5475	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩ ∈ V
46	44, 13, 45	fvmpt 7016	. . . . . . . 8 ⊢ (𝑤 ∈ (1...𝑁) → (𝑇‘𝑤) = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩)
47	41, 46	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑇‘𝑤) = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩)
48	40, 47	eqeq12d 2751	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝑇‘𝑧) = (𝑇‘𝑤) ↔ ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩))
49		fvex 6920	. . . . . . . 8 ⊢ (𝐼‘𝑧) ∈ V
50		fvex 6920	. . . . . . . 8 ⊢ (𝐽‘𝑧) ∈ V
51	49, 50	opth 5487	. . . . . . 7 ⊢ (⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩ ↔ ((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤)))
52	34, 30	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑧 ∈ ℝ)
53	31, 41	sselid 3993	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑤 ∈ ℝ)
54		simpr3 1195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑧 ≤ 𝑤)
55	52, 53, 54	leltned 11412	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑧 < 𝑤 ↔ 𝑤 ≠ 𝑧))
56	2	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝐹:(1...𝑁)–1-1→ℝ)
57		f1fveq 7282	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(1...𝑁)–1-1→ℝ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
58	56, 34, 41, 57	syl12anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
59	58, 26	bitr4di 289	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑤 = 𝑧))
60	59	necon3bid 2983	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) ↔ 𝑤 ≠ 𝑧))
61	55, 60	bitr4d 282	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑧 < 𝑤 ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
62	61	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (𝐹‘𝑧) ≠ (𝐹‘𝑤))
63		f1f 6805	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:(1...𝑁)–1-1→ℝ → 𝐹:(1...𝑁)⟶ℝ)
64	2, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:(1...𝑁)⟶ℝ)
65	64	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)⟶ℝ)
66	34	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (1...𝑁))
67	65, 66	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (𝐹‘𝑧) ∈ ℝ)
68	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (1...𝑁))
69	65, 68	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (𝐹‘𝑤) ∈ ℝ)
70	67, 69	lttri2d 11398	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) ↔ ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧))))
71	62, 70	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧)))
72	1	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑁 ∈ ℕ)
73	2	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)–1-1→ℝ)
74		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
75	72, 73, 3, 4, 66, 68, 74	erdszelem8 35183	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ((𝐼‘𝑧) = (𝐼‘𝑤) → ¬ (𝐹‘𝑧) < (𝐹‘𝑤)))
76	72, 73, 7, 8, 66, 68, 74	erdszelem8 35183	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ((𝐽‘𝑧) = (𝐽‘𝑤) → ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤)))
77	75, 76	anim12d 609	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤)) → (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤))))
78		ioran 985	. . . . . . . . . . . . 13 ⊢ (¬ ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧)) ↔ (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑤) < (𝐹‘𝑧)))
79		fvex 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑧) ∈ V
80		fvex 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑤) ∈ V
81	79, 80	brcnv 5896	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑧)◡ < (𝐹‘𝑤) ↔ (𝐹‘𝑤) < (𝐹‘𝑧))
82	81	notbii 320	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤) ↔ ¬ (𝐹‘𝑤) < (𝐹‘𝑧))
83	82	anbi2i 623	. . . . . . . . . . . . 13 ⊢ ((¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤)) ↔ (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑤) < (𝐹‘𝑧)))
84	78, 83	bitr4i 278	. . . . . . . . . . . 12 ⊢ (¬ ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧)) ↔ (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤)))
85	77, 84	imbitrrdi 252	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤)) → ¬ ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧))))
86	71, 85	mt2d 136	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ¬ ((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤)))
87	86	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑧 < 𝑤 → ¬ ((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤))))
88	55, 87	sylbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑤 ≠ 𝑧 → ¬ ((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤))))
89	88	necon4ad 2957	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤)) → 𝑤 = 𝑧))
90	51, 89	biimtrid 242	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩ → 𝑤 = 𝑧))
91	48, 90	sylbid 240	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑤 = 𝑧))
92	91, 26	imbitrdi 251	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
93	19, 28, 32, 33, 92	wlogle 11794	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
94	93	ralrimivva 3200	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
95		dff13 7275	. 2 ⊢ (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) ↔ (𝑇:(1...𝑁)⟶(ℕ × ℕ) ∧ ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤)))
96	14, 94, 95	sylanbrc 583	1 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  {crab 3433   ⊆ wss 3963  𝒫 cpw 4605  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231   × cxp 5687  ^◡ccnv 5688   ↾ cres 5691   “ cima 5692  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563   Isom wiso 6564  (class class class)co 7431  supcsup 9478  ℝcr 11152  1c1 11154   < clt 11293   ≤ cle 11294  ℕcn 12264  ...cfz 13544  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367

This theorem is referenced by:  erdszelem10  35185