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Theorem erdszelem9 32439
Description: Lemma for erdsze 32442. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n (𝜑𝑁 ∈ ℕ)
erdsze.f (𝜑𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
erdszelem.j 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
erdszelem.t 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
Assertion
Ref Expression
erdszelem9 (𝜑𝑇:(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.
StepHypRef Expression
1 erdsze.n . . . . . 6 (𝜑𝑁 ∈ ℕ)
2 erdsze.f . . . . . 6 (𝜑𝐹:(1...𝑁)–1-1→ℝ)
3 erdszelem.i . . . . . 6 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
4 ltso 10713 . . . . . 6 < Or ℝ
51, 2, 3, 4erdszelem6 32436 . . . . 5 (𝜑𝐼:(1...𝑁)⟶ℕ)
65ffvelrnda 6844 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐼𝑛) ∈ ℕ)
7 erdszelem.j . . . . . 6 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
8 gtso 10714 . . . . . 6 < Or ℝ
91, 2, 7, 8erdszelem6 32436 . . . . 5 (𝜑𝐽:(1...𝑁)⟶ℕ)
109ffvelrnda 6844 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐽𝑛) ∈ ℕ)
11 opelxpi 5585 . . . 4 (((𝐼𝑛) ∈ ℕ ∧ (𝐽𝑛) ∈ ℕ) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
126, 10, 11syl2anc 586 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
13 erdszelem.t . . 3 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
1412, 13fmptd 6871 . 2 (𝜑𝑇:(1...𝑁)⟶(ℕ × ℕ))
15 fveq2 6663 . . . . . 6 (𝑎 = 𝑧 → (𝑇𝑎) = (𝑇𝑧))
16 fveq2 6663 . . . . . 6 (𝑏 = 𝑤 → (𝑇𝑏) = (𝑇𝑤))
1715, 16eqeqan12d 2836 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
18 eqeq12 2833 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → (𝑎 = 𝑏𝑧 = 𝑤))
1917, 18imbi12d 347 . . . 4 ((𝑎 = 𝑧𝑏 = 𝑤) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
20 fveq2 6663 . . . . . . 7 (𝑎 = 𝑤 → (𝑇𝑎) = (𝑇𝑤))
21 fveq2 6663 . . . . . . 7 (𝑏 = 𝑧 → (𝑇𝑏) = (𝑇𝑧))
2220, 21eqeqan12d 2836 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑤) = (𝑇𝑧)))
23 eqcom 2826 . . . . . 6 ((𝑇𝑤) = (𝑇𝑧) ↔ (𝑇𝑧) = (𝑇𝑤))
2422, 23syl6bb 289 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
25 eqeq12 2833 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑤 = 𝑧))
26 eqcom 2826 . . . . . 6 (𝑤 = 𝑧𝑧 = 𝑤)
2725, 26syl6bb 289 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑧 = 𝑤))
2824, 27imbi12d 347 . . . 4 ((𝑎 = 𝑤𝑏 = 𝑧) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
29 elfzelz 12900 . . . . . . 7 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3029zred 12079 . . . . . 6 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
3130ssriv 3969 . . . . 5 (1...𝑁) ⊆ ℝ
3231a1i 11 . . . 4 (𝜑 → (1...𝑁) ⊆ ℝ)
33 biidd 264 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → (((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
34 simpr1 1189 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧 ∈ (1...𝑁))
35 fveq2 6663 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐼𝑛) = (𝐼𝑧))
36 fveq2 6663 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐽𝑛) = (𝐽𝑧))
3735, 36opeq12d 4803 . . . . . . . . 9 (𝑛 = 𝑧 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
38 opex 5347 . . . . . . . . 9 ⟨(𝐼𝑧), (𝐽𝑧)⟩ ∈ V
3937, 13, 38fvmpt 6761 . . . . . . . 8 (𝑧 ∈ (1...𝑁) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
4034, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
41 simpr2 1190 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ (1...𝑁))
42 fveq2 6663 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐼𝑛) = (𝐼𝑤))
43 fveq2 6663 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐽𝑛) = (𝐽𝑤))
4442, 43opeq12d 4803 . . . . . . . . 9 (𝑛 = 𝑤 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
45 opex 5347 . . . . . . . . 9 ⟨(𝐼𝑤), (𝐽𝑤)⟩ ∈ V
4644, 13, 45fvmpt 6761 . . . . . . . 8 (𝑤 ∈ (1...𝑁) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4741, 46syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4840, 47eqeq12d 2835 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) ↔ ⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩))
49 fvex 6676 . . . . . . . 8 (𝐼𝑧) ∈ V
50 fvex 6676 . . . . . . . 8 (𝐽𝑧) ∈ V
5149, 50opth 5359 . . . . . . 7 (⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩ ↔ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)))
5234, 30syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧 ∈ ℝ)
5331, 41sseldi 3963 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ ℝ)
54 simpr3 1191 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧𝑤)
5552, 53, 54leltned 10785 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤𝑤𝑧))
562adantr 483 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝐹:(1...𝑁)–1-1→ℝ)
57 f1fveq 7012 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)–1-1→ℝ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5856, 34, 41, 57syl12anc 834 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5958, 26syl6bbr 291 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑤 = 𝑧))
6059necon3bid 3058 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ 𝑤𝑧))
6155, 60bitr4d 284 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 ↔ (𝐹𝑧) ≠ (𝐹𝑤)))
6261biimpa 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ≠ (𝐹𝑤))
63 f1f 6568 . . . . . . . . . . . . . . . 16 (𝐹:(1...𝑁)–1-1→ℝ → 𝐹:(1...𝑁)⟶ℝ)
642, 63syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:(1...𝑁)⟶ℝ)
6564ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)⟶ℝ)
6634adantr 483 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (1...𝑁))
6765, 66ffvelrnd 6845 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ∈ ℝ)
6841adantr 483 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (1...𝑁))
6965, 68ffvelrnd 6845 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑤) ∈ ℝ)
7067, 69lttri2d 10771 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧))))
7162, 70mpbid 234 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)))
721ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑁 ∈ ℕ)
732ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)–1-1→ℝ)
74 simpr 487 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
7572, 73, 3, 4, 66, 68, 74erdszelem8 32438 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐼𝑧) = (𝐼𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7672, 73, 7, 8, 66, 68, 74erdszelem8 32438 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐽𝑧) = (𝐽𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7775, 76anim12d 610 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤))))
78 ioran 980 . . . . . . . . . . . . 13 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
79 fvex 6676 . . . . . . . . . . . . . . . 16 (𝐹𝑧) ∈ V
80 fvex 6676 . . . . . . . . . . . . . . . 16 (𝐹𝑤) ∈ V
8179, 80brcnv 5746 . . . . . . . . . . . . . . 15 ((𝐹𝑧) < (𝐹𝑤) ↔ (𝐹𝑤) < (𝐹𝑧))
8281notbii 322 . . . . . . . . . . . . . 14 (¬ (𝐹𝑧) < (𝐹𝑤) ↔ ¬ (𝐹𝑤) < (𝐹𝑧))
8382anbi2i 624 . . . . . . . . . . . . 13 ((¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
8478, 83bitr4i 280 . . . . . . . . . . . 12 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)))
8577, 84syl6ibr 254 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → ¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧))))
8671, 85mt2d 138 . . . . . . . . . 10 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)))
8786ex 415 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8855, 87sylbird 262 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑤𝑧 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8988necon4ad 3033 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → 𝑤 = 𝑧))
9051, 89syl5bi 244 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩ → 𝑤 = 𝑧))
9148, 90sylbid 242 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑤 = 𝑧))
9291, 26syl6ib 253 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9319, 28, 32, 33, 92wlogle 11165 . . 3 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9493ralrimivva 3189 . 2 (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
95 dff13 7005 . 2 (𝑇:(1...𝑁)–1-1→(ℕ × ℕ) ↔ (𝑇:(1...𝑁)⟶(ℕ × ℕ) ∧ ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
9614, 94, 95sylanbrc 585 1 (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1082   = wceq 1531  wcel 2108  wne 3014  wral 3136  {crab 3140  wss 3934  𝒫 cpw 4537  cop 4565   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  cres 5550  cima 5551  wf 6344  1-1wf1 6345  cfv 6348   Isom wiso 6349  (class class class)co 7148  supcsup 8896  cr 10528  1c1 10530   < clt 10667  cle 10668  cn 11630  ...cfz 12884  chash 13682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12885  df-hash 13683
This theorem is referenced by:  erdszelem10  32440
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