Theorem erdszelem9 35167

Description: Lemma for erdsze 35170. (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 11370	. . . . . 6 ⊢ < Or ℝ
5	1, 2, 3, 4	erdszelem6 35164	. . . . 5 ⊢ (𝜑 → 𝐼:(1...𝑁)⟶ℕ)
6	5	ffvelcdmda 7118	. . . 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 11371	. . . . . 6 ⊢ ◡ < Or ℝ
9	1, 2, 7, 8	erdszelem6 35164	. . . . 5 ⊢ (𝜑 → 𝐽:(1...𝑁)⟶ℕ)
10	9	ffvelcdmda 7118	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝐽‘𝑛) ∈ ℕ)
11		opelxpi 5737	. . . 4 ⊢ (((𝐼‘𝑛) ∈ ℕ ∧ (𝐽‘𝑛) ∈ ℕ) → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ ∈ (ℕ × ℕ))
12	6, 10, 11	syl2anc 583	. . 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 7148	. 2 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(ℕ × ℕ))
15		fveq2 6920	. . . . . 6 ⊢ (𝑎 = 𝑧 → (𝑇‘𝑎) = (𝑇‘𝑧))
16		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝑤 → (𝑇‘𝑏) = (𝑇‘𝑤))
17	15, 16	eqeqan12d 2754	. . . . 5 ⊢ ((𝑎 = 𝑧 ∧ 𝑏 = 𝑤) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑧) = (𝑇‘𝑤)))
18		eqeq12 2757	. . . . 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 6920	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑇‘𝑎) = (𝑇‘𝑤))
21		fveq2 6920	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (𝑇‘𝑏) = (𝑇‘𝑧))
22	20, 21	eqeqan12d 2754	. . . . . 6 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑤) = (𝑇‘𝑧)))
23		eqcom 2747	. . . . . 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 2757	. . . . . 6 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → (𝑎 = 𝑏 ↔ 𝑤 = 𝑧))
26		eqcom 2747	. . . . . 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 13584	. . . . . . 7 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
30	29	zred 12747	. . . . . 6 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
31	30	ssriv 4012	. . . . 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 1194	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑧 ∈ (1...𝑁))
35		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝐼‘𝑛) = (𝐼‘𝑧))
36		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝐽‘𝑛) = (𝐽‘𝑧))
37	35, 36	opeq12d 4905	. . . . . . . . 9 ⊢ (𝑛 = 𝑧 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩)
38		opex 5484	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ ∈ V
39	37, 13, 38	fvmpt 7029	. . . . . . . 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 1195	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑤 ∈ (1...𝑁))
42		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑛 = 𝑤 → (𝐼‘𝑛) = (𝐼‘𝑤))
43		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑛 = 𝑤 → (𝐽‘𝑛) = (𝐽‘𝑤))
44	42, 43	opeq12d 4905	. . . . . . . . 9 ⊢ (𝑛 = 𝑤 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩)
45		opex 5484	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩ ∈ V
46	44, 13, 45	fvmpt 7029	. . . . . . . 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 2756	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝑇‘𝑧) = (𝑇‘𝑤) ↔ ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩))
49		fvex 6933	. . . . . . . 8 ⊢ (𝐼‘𝑧) ∈ V
50		fvex 6933	. . . . . . . 8 ⊢ (𝐽‘𝑧) ∈ V
51	49, 50	opth 5496	. . . . . . 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 4006	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑤 ∈ ℝ)
54		simpr3 1196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑧 ≤ 𝑤)
55	52, 53, 54	leltned 11443	. . . . . . . . 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 7299	. . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . 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 2991	. . . . . . . . . . . . . 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 6817	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . 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 7119	. . . . . . . . . . . . 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 7119	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (𝐹‘𝑤) ∈ ℝ)
70	67, 69	lttri2d 11429	. . . . . . . . . . . 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 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑁 ∈ ℕ)
73	2	ad2antrr 725	. . . . . . . . . . . . . 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 35166	. . . . . . . . . . . . 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 35166	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ((𝐽‘𝑧) = (𝐽‘𝑤) → ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤)))
77	75, 76	anim12d 608	. . . . . . . . . . . 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 984	. . . . . . . . . . . . 13 ⊢ (¬ ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧)) ↔ (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑤) < (𝐹‘𝑧)))
79		fvex 6933	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑧) ∈ V
80		fvex 6933	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑤) ∈ V
81	79, 80	brcnv 5907	. . . . . . . . . . . . . . 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 622	. . . . . . . . . . . . 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 2965	. . . . . . 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 11823	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
94	93	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
95		dff13 7292	. 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 582	1 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443   ⊆ wss 3976  𝒫 cpw 4622  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699   ↾ cres 5702   “ cima 5703  ⟶wf 6569  –1-1→wf1 6570  ‘cfv 6573   Isom wiso 6574  (class class class)co 7448  supcsup 9509  ℝcr 11183  1c1 11185   < clt 11324   ≤ cle 11325  ℕcn 12293  ...cfz 13567  ♯chash 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380

This theorem is referenced by:  erdszelem10  35168