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Theorem erdszelem9 30881
Description: Lemma for erdsze 30884. (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 10063 . . . . . 6 < Or ℝ
51, 2, 3, 4erdszelem6 30878 . . . . 5 (𝜑𝐼:(1...𝑁)⟶ℕ)
65ffvelrnda 6316 . . . 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 10064 . . . . . 6 < Or ℝ
91, 2, 7, 8erdszelem6 30878 . . . . 5 (𝜑𝐽:(1...𝑁)⟶ℕ)
109ffvelrnda 6316 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐽𝑛) ∈ ℕ)
11 opelxpi 5113 . . . 4 (((𝐼𝑛) ∈ ℕ ∧ (𝐽𝑛) ∈ ℕ) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
126, 10, 11syl2anc 692 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
13 erdszelem.t . . 3 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
1412, 13fmptd 6341 . 2 (𝜑𝑇:(1...𝑁)⟶(ℕ × ℕ))
15 fveq2 6150 . . . . . 6 (𝑎 = 𝑧 → (𝑇𝑎) = (𝑇𝑧))
16 fveq2 6150 . . . . . 6 (𝑏 = 𝑤 → (𝑇𝑏) = (𝑇𝑤))
1715, 16eqeqan12d 2642 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
18 eqeq12 2639 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → (𝑎 = 𝑏𝑧 = 𝑤))
1917, 18imbi12d 334 . . . 4 ((𝑎 = 𝑧𝑏 = 𝑤) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
20 fveq2 6150 . . . . . . 7 (𝑎 = 𝑤 → (𝑇𝑎) = (𝑇𝑤))
21 fveq2 6150 . . . . . . 7 (𝑏 = 𝑧 → (𝑇𝑏) = (𝑇𝑧))
2220, 21eqeqan12d 2642 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑤) = (𝑇𝑧)))
23 eqcom 2633 . . . . . 6 ((𝑇𝑤) = (𝑇𝑧) ↔ (𝑇𝑧) = (𝑇𝑤))
2422, 23syl6bb 276 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
25 eqeq12 2639 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑤 = 𝑧))
26 eqcom 2633 . . . . . 6 (𝑤 = 𝑧𝑧 = 𝑤)
2725, 26syl6bb 276 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑧 = 𝑤))
2824, 27imbi12d 334 . . . 4 ((𝑎 = 𝑤𝑏 = 𝑧) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
29 elfzelz 12281 . . . . . . 7 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3029zred 11426 . . . . . 6 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
3130ssriv 3592 . . . . 5 (1...𝑁) ⊆ ℝ
3231a1i 11 . . . 4 (𝜑 → (1...𝑁) ⊆ ℝ)
33 biidd 252 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → (((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
34 simpr1 1065 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧 ∈ (1...𝑁))
35 fveq2 6150 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐼𝑛) = (𝐼𝑧))
36 fveq2 6150 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐽𝑛) = (𝐽𝑧))
3735, 36opeq12d 4383 . . . . . . . . 9 (𝑛 = 𝑧 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
38 opex 4898 . . . . . . . . 9 ⟨(𝐼𝑧), (𝐽𝑧)⟩ ∈ V
3937, 13, 38fvmpt 6240 . . . . . . . 8 (𝑧 ∈ (1...𝑁) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
4034, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
41 simpr2 1066 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ (1...𝑁))
42 fveq2 6150 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐼𝑛) = (𝐼𝑤))
43 fveq2 6150 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐽𝑛) = (𝐽𝑤))
4442, 43opeq12d 4383 . . . . . . . . 9 (𝑛 = 𝑤 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
45 opex 4898 . . . . . . . . 9 ⟨(𝐼𝑤), (𝐽𝑤)⟩ ∈ V
4644, 13, 45fvmpt 6240 . . . . . . . 8 (𝑤 ∈ (1...𝑁) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4741, 46syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4840, 47eqeq12d 2641 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) ↔ ⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩))
49 fvex 6160 . . . . . . . 8 (𝐼𝑧) ∈ V
50 fvex 6160 . . . . . . . 8 (𝐽𝑧) ∈ V
5149, 50opth 4910 . . . . . . 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 3586 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ ℝ)
54 simpr3 1067 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧𝑤)
5552, 53, 54leltned 10135 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤𝑤𝑧))
562adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝐹:(1...𝑁)–1-1→ℝ)
57 f1fveq 6474 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)–1-1→ℝ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5856, 34, 41, 57syl12anc 1321 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5958, 26syl6bbr 278 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑤 = 𝑧))
6059necon3bid 2840 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ 𝑤𝑧))
6155, 60bitr4d 271 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 ↔ (𝐹𝑧) ≠ (𝐹𝑤)))
6261biimpa 501 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ≠ (𝐹𝑤))
63 f1f 6060 . . . . . . . . . . . . . . . 16 (𝐹:(1...𝑁)–1-1→ℝ → 𝐹:(1...𝑁)⟶ℝ)
642, 63syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:(1...𝑁)⟶ℝ)
6564ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)⟶ℝ)
6634adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (1...𝑁))
6765, 66ffvelrnd 6317 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ∈ ℝ)
6841adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (1...𝑁))
6965, 68ffvelrnd 6317 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑤) ∈ ℝ)
7067, 69lttri2d 10121 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧))))
7162, 70mpbid 222 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)))
721ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑁 ∈ ℕ)
732ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)–1-1→ℝ)
74 simpr 477 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
7572, 73, 3, 4, 66, 68, 74erdszelem8 30880 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐼𝑧) = (𝐼𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7672, 73, 7, 8, 66, 68, 74erdszelem8 30880 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐽𝑧) = (𝐽𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7775, 76anim12d 585 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤))))
78 ioran 511 . . . . . . . . . . . . 13 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
79 fvex 6160 . . . . . . . . . . . . . . . 16 (𝐹𝑧) ∈ V
80 fvex 6160 . . . . . . . . . . . . . . . 16 (𝐹𝑤) ∈ V
8179, 80brcnv 5270 . . . . . . . . . . . . . . 15 ((𝐹𝑧) < (𝐹𝑤) ↔ (𝐹𝑤) < (𝐹𝑧))
8281notbii 310 . . . . . . . . . . . . . 14 (¬ (𝐹𝑧) < (𝐹𝑤) ↔ ¬ (𝐹𝑤) < (𝐹𝑧))
8382anbi2i 729 . . . . . . . . . . . . 13 ((¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
8478, 83bitr4i 267 . . . . . . . . . . . 12 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)))
8577, 84syl6ibr 242 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → ¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧))))
8671, 85mt2d 131 . . . . . . . . . 10 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)))
8786ex 450 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8855, 87sylbird 250 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑤𝑧 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8988necon4ad 2815 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → 𝑤 = 𝑧))
9051, 89syl5bi 232 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩ → 𝑤 = 𝑧))
9148, 90sylbid 230 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑤 = 𝑧))
9291, 26syl6ib 241 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9319, 28, 32, 33, 92wlogle 10506 . . 3 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9493ralrimivva 2970 . 2 (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
95 dff13 6467 . 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 697 1 (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  {crab 2916  wss 3560  𝒫 cpw 4135  cop 4159   class class class wbr 4618  cmpt 4678   × cxp 5077  ccnv 5078  cres 5081  cima 5082  wf 5846  1-1wf1 5847  cfv 5850   Isom wiso 5851  (class class class)co 6605  supcsup 8291  cr 9880  1c1 9882   < clt 10019  cle 10020  cn 10965  ...cfz 12265  #chash 13054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055
This theorem is referenced by:  erdszelem10  30882
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