Theorem erdszelem9 31181

Description: Lemma for erdsze 31184. (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 10118	. . . . . 6 ⊢ < Or ℝ
5	1, 2, 3, 4	erdszelem6 31178	. . . . 5 ⊢ (𝜑 → 𝐼:(1...𝑁)⟶ℕ)
6	5	ffvelrnda 6359	. . . 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 10119	. . . . . 6 ⊢ ◡ < Or ℝ
9	1, 2, 7, 8	erdszelem6 31178	. . . . 5 ⊢ (𝜑 → 𝐽:(1...𝑁)⟶ℕ)
10	9	ffvelrnda 6359	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝐽‘𝑛) ∈ ℕ)
11		opelxpi 5148	. . . 4 ⊢ (((𝐼‘𝑛) ∈ ℕ ∧ (𝐽‘𝑛) ∈ ℕ) → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ ∈ (ℕ × ℕ))
12	6, 10, 11	syl2anc 693	. . 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 6385	. 2 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(ℕ × ℕ))
15		fveq2 6191	. . . . . 6 ⊢ (𝑎 = 𝑧 → (𝑇‘𝑎) = (𝑇‘𝑧))
16		fveq2 6191	. . . . . 6 ⊢ (𝑏 = 𝑤 → (𝑇‘𝑏) = (𝑇‘𝑤))
17	15, 16	eqeqan12d 2638	. . . . 5 ⊢ ((𝑎 = 𝑧 ∧ 𝑏 = 𝑤) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑧) = (𝑇‘𝑤)))
18		eqeq12 2635	. . . . 5 ⊢ ((𝑎 = 𝑧 ∧ 𝑏 = 𝑤) → (𝑎 = 𝑏 ↔ 𝑧 = 𝑤))
19	17, 18	imbi12d 334	. . . 4 ⊢ ((𝑎 = 𝑧 ∧ 𝑏 = 𝑤) → (((𝑇‘𝑎) = (𝑇‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤)))
20		fveq2 6191	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑇‘𝑎) = (𝑇‘𝑤))
21		fveq2 6191	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (𝑇‘𝑏) = (𝑇‘𝑧))
22	20, 21	eqeqan12d 2638	. . . . . 6 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑤) = (𝑇‘𝑧)))
23		eqcom 2629	. . . . . 6 ⊢ ((𝑇‘𝑤) = (𝑇‘𝑧) ↔ (𝑇‘𝑧) = (𝑇‘𝑤))
24	22, 23	syl6bb 276	. . . . 5 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → ((𝑇‘𝑎) = (𝑇‘𝑏) ↔ (𝑇‘𝑧) = (𝑇‘𝑤)))
25		eqeq12 2635	. . . . . 6 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → (𝑎 = 𝑏 ↔ 𝑤 = 𝑧))
26		eqcom 2629	. . . . . 6 ⊢ (𝑤 = 𝑧 ↔ 𝑧 = 𝑤)
27	25, 26	syl6bb 276	. . . . 5 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → (𝑎 = 𝑏 ↔ 𝑧 = 𝑤))
28	24, 27	imbi12d 334	. . . 4 ⊢ ((𝑎 = 𝑤 ∧ 𝑏 = 𝑧) → (((𝑇‘𝑎) = (𝑇‘𝑏) → 𝑎 = 𝑏) ↔ ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤)))
29		elfzelz 12342	. . . . . . 7 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
30	29	zred 11482	. . . . . 6 ⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
31	30	ssriv 3607	. . . . 5 ⊢ (1...𝑁) ⊆ ℝ
32	31	a1i 11	. . . 4 ⊢ (𝜑 → (1...𝑁) ⊆ ℝ)
33		biidd 252	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → (((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤) ↔ ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤)))
34		simpr1 1067	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑧 ∈ (1...𝑁))
35		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝐼‘𝑛) = (𝐼‘𝑧))
36		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → (𝐽‘𝑛) = (𝐽‘𝑧))
37	35, 36	opeq12d 4410	. . . . . . . . 9 ⊢ (𝑛 = 𝑧 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩)
38		opex 4932	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ ∈ V
39	37, 13, 38	fvmpt 6282	. . . . . . . 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 1068	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑤 ∈ (1...𝑁))
42		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑛 = 𝑤 → (𝐼‘𝑛) = (𝐼‘𝑤))
43		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑛 = 𝑤 → (𝐽‘𝑛) = (𝐽‘𝑤))
44	42, 43	opeq12d 4410	. . . . . . . . 9 ⊢ (𝑛 = 𝑤 → ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩)
45		opex 4932	. . . . . . . . 9 ⊢ ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩ ∈ V
46	44, 13, 45	fvmpt 6282	. . . . . . . 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 2637	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝑇‘𝑧) = (𝑇‘𝑤) ↔ ⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩))
49		fvex 6201	. . . . . . . 8 ⊢ (𝐼‘𝑧) ∈ V
50		fvex 6201	. . . . . . . 8 ⊢ (𝐽‘𝑧) ∈ V
51	49, 50	opth 4945	. . . . . . 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	sseldi 3601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑤 ∈ ℝ)
54		simpr3 1069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝑧 ≤ 𝑤)
55	52, 53, 54	leltned 10190	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑧 < 𝑤 ↔ 𝑤 ≠ 𝑧))
56	2	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → 𝐹:(1...𝑁)–1-1→ℝ)
57		f1fveq 6519	. . . . . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
59	58, 26	syl6bbr 278	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑤 = 𝑧))
60	59	necon3bid 2838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝐹‘𝑧) ≠ (𝐹‘𝑤) ↔ 𝑤 ≠ 𝑧))
61	55, 60	bitr4d 271	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑧 < 𝑤 ↔ (𝐹‘𝑧) ≠ (𝐹‘𝑤)))
62	61	biimpa 501	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (𝐹‘𝑧) ≠ (𝐹‘𝑤))
63		f1f 6101	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)⟶ℝ)
66	34	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (1...𝑁))
67	65, 66	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (𝐹‘𝑧) ∈ ℝ)
68	41	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (1...𝑁))
69	65, 68	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → (𝐹‘𝑤) ∈ ℝ)
70	67, 69	lttri2d 10176	. . . . . . . . . . . 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 222	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧)))
72	1	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑁 ∈ ℕ)
73	2	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)–1-1→ℝ)
74		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
75	72, 73, 3, 4, 66, 68, 74	erdszelem8 31180	. . . . . . . . . . . . 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 31180	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ((𝐽‘𝑧) = (𝐽‘𝑤) → ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤)))
77	75, 76	anim12d 586	. . . . . . . . . . . 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 511	. . . . . . . . . . . . 13 ⊢ (¬ ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧)) ↔ (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑤) < (𝐹‘𝑧)))
79		fvex 6201	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑧) ∈ V
80		fvex 6201	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑤) ∈ V
81	79, 80	brcnv 5305	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑧)◡ < (𝐹‘𝑤) ↔ (𝐹‘𝑤) < (𝐹‘𝑧))
82	81	notbii 310	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤) ↔ ¬ (𝐹‘𝑤) < (𝐹‘𝑧))
83	82	anbi2i 730	. . . . . . . . . . . . 13 ⊢ ((¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤)) ↔ (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑤) < (𝐹‘𝑧)))
84	78, 83	bitr4i 267	. . . . . . . . . . . 12 ⊢ (¬ ((𝐹‘𝑧) < (𝐹‘𝑤) ∨ (𝐹‘𝑤) < (𝐹‘𝑧)) ↔ (¬ (𝐹‘𝑧) < (𝐹‘𝑤) ∧ ¬ (𝐹‘𝑧)◡ < (𝐹‘𝑤)))
85	77, 84	syl6ibr 242	. . . . . . . . . . 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 131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) ∧ 𝑧 < 𝑤) → ¬ ((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤)))
87	86	ex 450	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑧 < 𝑤 → ¬ ((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤))))
88	55, 87	sylbird 250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (𝑤 ≠ 𝑧 → ¬ ((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤))))
89	88	necon4ad 2813	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (((𝐼‘𝑧) = (𝐼‘𝑤) ∧ (𝐽‘𝑧) = (𝐽‘𝑤)) → 𝑤 = 𝑧))
90	51, 89	syl5bi 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → (⟨(𝐼‘𝑧), (𝐽‘𝑧)⟩ = ⟨(𝐼‘𝑤), (𝐽‘𝑤)⟩ → 𝑤 = 𝑧))
91	48, 90	sylbid 230	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑤 = 𝑧))
92	91, 26	syl6ib 241	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧 ≤ 𝑤)) → ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
93	19, 28, 32, 33, 92	wlogle 10561	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
94	93	ralrimivva 2971	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇‘𝑧) = (𝑇‘𝑤) → 𝑧 = 𝑤))
95		dff13 6512	. 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 698	1 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   “ cima 5117  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650  supcsup 8346  ℝcr 9935  1c1 9937   < clt 10074   ≤ cle 10075  ℕcn 11020  ...cfz 12326  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  erdszelem10  31182