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Theorem erdszelem9 32667
 Description: Lemma for erdsze 32670. (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 10749 . . . . . 6 < Or ℝ
51, 2, 3, 4erdszelem6 32664 . . . . 5 (𝜑𝐼:(1...𝑁)⟶ℕ)
65ffvelrnda 6840 . . . 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 10750 . . . . . 6 < Or ℝ
91, 2, 7, 8erdszelem6 32664 . . . . 5 (𝜑𝐽:(1...𝑁)⟶ℕ)
109ffvelrnda 6840 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐽𝑛) ∈ ℕ)
11 opelxpi 5559 . . . 4 (((𝐼𝑛) ∈ ℕ ∧ (𝐽𝑛) ∈ ℕ) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
126, 10, 11syl2anc 588 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
13 erdszelem.t . . 3 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
1412, 13fmptd 6867 . 2 (𝜑𝑇:(1...𝑁)⟶(ℕ × ℕ))
15 fveq2 6656 . . . . . 6 (𝑎 = 𝑧 → (𝑇𝑎) = (𝑇𝑧))
16 fveq2 6656 . . . . . 6 (𝑏 = 𝑤 → (𝑇𝑏) = (𝑇𝑤))
1715, 16eqeqan12d 2776 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
18 eqeq12 2773 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → (𝑎 = 𝑏𝑧 = 𝑤))
1917, 18imbi12d 349 . . . 4 ((𝑎 = 𝑧𝑏 = 𝑤) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
20 fveq2 6656 . . . . . . 7 (𝑎 = 𝑤 → (𝑇𝑎) = (𝑇𝑤))
21 fveq2 6656 . . . . . . 7 (𝑏 = 𝑧 → (𝑇𝑏) = (𝑇𝑧))
2220, 21eqeqan12d 2776 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑤) = (𝑇𝑧)))
23 eqcom 2766 . . . . . 6 ((𝑇𝑤) = (𝑇𝑧) ↔ (𝑇𝑧) = (𝑇𝑤))
2422, 23syl6bb 291 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
25 eqeq12 2773 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑤 = 𝑧))
26 eqcom 2766 . . . . . 6 (𝑤 = 𝑧𝑧 = 𝑤)
2725, 26syl6bb 291 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑧 = 𝑤))
2824, 27imbi12d 349 . . . 4 ((𝑎 = 𝑤𝑏 = 𝑧) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
29 elfzelz 12946 . . . . . . 7 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3029zred 12116 . . . . . 6 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
3130ssriv 3897 . . . . 5 (1...𝑁) ⊆ ℝ
3231a1i 11 . . . 4 (𝜑 → (1...𝑁) ⊆ ℝ)
33 biidd 265 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → (((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
34 simpr1 1192 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧 ∈ (1...𝑁))
35 fveq2 6656 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐼𝑛) = (𝐼𝑧))
36 fveq2 6656 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐽𝑛) = (𝐽𝑧))
3735, 36opeq12d 4769 . . . . . . . . 9 (𝑛 = 𝑧 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
38 opex 5322 . . . . . . . . 9 ⟨(𝐼𝑧), (𝐽𝑧)⟩ ∈ V
3937, 13, 38fvmpt 6757 . . . . . . . 8 (𝑧 ∈ (1...𝑁) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
4034, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
41 simpr2 1193 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ (1...𝑁))
42 fveq2 6656 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐼𝑛) = (𝐼𝑤))
43 fveq2 6656 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐽𝑛) = (𝐽𝑤))
4442, 43opeq12d 4769 . . . . . . . . 9 (𝑛 = 𝑤 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
45 opex 5322 . . . . . . . . 9 ⟨(𝐼𝑤), (𝐽𝑤)⟩ ∈ V
4644, 13, 45fvmpt 6757 . . . . . . . 8 (𝑤 ∈ (1...𝑁) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4741, 46syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4840, 47eqeq12d 2775 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) ↔ ⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩))
49 fvex 6669 . . . . . . . 8 (𝐼𝑧) ∈ V
50 fvex 6669 . . . . . . . 8 (𝐽𝑧) ∈ V
5149, 50opth 5334 . . . . . . 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 3891 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ ℝ)
54 simpr3 1194 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧𝑤)
5552, 53, 54leltned 10821 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤𝑤𝑧))
562adantr 485 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝐹:(1...𝑁)–1-1→ℝ)
57 f1fveq 7010 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)–1-1→ℝ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5856, 34, 41, 57syl12anc 836 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5958, 26bitr4di 293 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑤 = 𝑧))
6059necon3bid 2996 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ 𝑤𝑧))
6155, 60bitr4d 285 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 ↔ (𝐹𝑧) ≠ (𝐹𝑤)))
6261biimpa 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ≠ (𝐹𝑤))
63 f1f 6558 . . . . . . . . . . . . . . . 16 (𝐹:(1...𝑁)–1-1→ℝ → 𝐹:(1...𝑁)⟶ℝ)
642, 63syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:(1...𝑁)⟶ℝ)
6564ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)⟶ℝ)
6634adantr 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (1...𝑁))
6765, 66ffvelrnd 6841 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ∈ ℝ)
6841adantr 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (1...𝑁))
6965, 68ffvelrnd 6841 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑤) ∈ ℝ)
7067, 69lttri2d 10807 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧))))
7162, 70mpbid 235 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)))
721ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑁 ∈ ℕ)
732ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)–1-1→ℝ)
74 simpr 489 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
7572, 73, 3, 4, 66, 68, 74erdszelem8 32666 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐼𝑧) = (𝐼𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7672, 73, 7, 8, 66, 68, 74erdszelem8 32666 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐽𝑧) = (𝐽𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7775, 76anim12d 612 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤))))
78 ioran 982 . . . . . . . . . . . . 13 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
79 fvex 6669 . . . . . . . . . . . . . . . 16 (𝐹𝑧) ∈ V
80 fvex 6669 . . . . . . . . . . . . . . . 16 (𝐹𝑤) ∈ V
8179, 80brcnv 5720 . . . . . . . . . . . . . . 15 ((𝐹𝑧) < (𝐹𝑤) ↔ (𝐹𝑤) < (𝐹𝑧))
8281notbii 324 . . . . . . . . . . . . . 14 (¬ (𝐹𝑧) < (𝐹𝑤) ↔ ¬ (𝐹𝑤) < (𝐹𝑧))
8382anbi2i 626 . . . . . . . . . . . . 13 ((¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
8478, 83bitr4i 281 . . . . . . . . . . . 12 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)))
8577, 84syl6ibr 255 . . . . . . . . . . 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 417 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8855, 87sylbird 263 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑤𝑧 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8988necon4ad 2971 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → 𝑤 = 𝑧))
9051, 89syl5bi 245 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩ → 𝑤 = 𝑧))
9148, 90sylbid 243 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑤 = 𝑧))
9291, 26syl6ib 254 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9319, 28, 32, 33, 92wlogle 11201 . . 3 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9493ralrimivva 3121 . 2 (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
95 dff13 7003 . 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 587 1 (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  {crab 3075   ⊆ wss 3859  𝒫 cpw 4492  ⟨cop 4526   class class class wbr 5030   ↦ cmpt 5110   × cxp 5520  ◡ccnv 5521   ↾ cres 5524   “ cima 5525  ⟶wf 6329  –1-1→wf1 6330  ‘cfv 6333   Isom wiso 6334  (class class class)co 7148  supcsup 8927  ℝcr 10564  1c1 10566   < clt 10703   ≤ cle 10704  ℕcn 11664  ...cfz 12929  ♯chash 13730 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457  ax-cnex 10621  ax-resscn 10622  ax-1cn 10623  ax-icn 10624  ax-addcl 10625  ax-addrcl 10626  ax-mulcl 10627  ax-mulrcl 10628  ax-mulcom 10629  ax-addass 10630  ax-mulass 10631  ax-distr 10632  ax-i2m1 10633  ax-1ne0 10634  ax-1rid 10635  ax-rnegex 10636  ax-rrecex 10637  ax-cnre 10638  ax-pre-lttri 10639  ax-pre-lttrn 10640  ax-pre-ltadd 10641  ax-pre-mulgt0 10642  ax-pre-sup 10643 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-isom 6342  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-1st 7691  df-2nd 7692  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-1o 8110  df-2o 8111  df-oadd 8114  df-er 8297  df-map 8416  df-en 8526  df-dom 8527  df-sdom 8528  df-fin 8529  df-sup 8929  df-dju 9353  df-card 9391  df-pnf 10705  df-mnf 10706  df-xr 10707  df-ltxr 10708  df-le 10709  df-sub 10900  df-neg 10901  df-nn 11665  df-n0 11925  df-xnn0 11997  df-z 12011  df-uz 12273  df-fz 12930  df-hash 13731 This theorem is referenced by:  erdszelem10  32668
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