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Theorem erdszelem9 35037
Description: Lemma for erdsze 35040. (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 11332 . . . . . 6 < Or ℝ
51, 2, 3, 4erdszelem6 35034 . . . . 5 (𝜑𝐼:(1...𝑁)⟶ℕ)
65ffvelcdmda 7087 . . . 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 11333 . . . . . 6 < Or ℝ
91, 2, 7, 8erdszelem6 35034 . . . . 5 (𝜑𝐽:(1...𝑁)⟶ℕ)
109ffvelcdmda 7087 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐽𝑛) ∈ ℕ)
11 opelxpi 5709 . . . 4 (((𝐼𝑛) ∈ ℕ ∧ (𝐽𝑛) ∈ ℕ) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
126, 10, 11syl2anc 582 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
13 erdszelem.t . . 3 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
1412, 13fmptd 7117 . 2 (𝜑𝑇:(1...𝑁)⟶(ℕ × ℕ))
15 fveq2 6890 . . . . . 6 (𝑎 = 𝑧 → (𝑇𝑎) = (𝑇𝑧))
16 fveq2 6890 . . . . . 6 (𝑏 = 𝑤 → (𝑇𝑏) = (𝑇𝑤))
1715, 16eqeqan12d 2740 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
18 eqeq12 2743 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → (𝑎 = 𝑏𝑧 = 𝑤))
1917, 18imbi12d 343 . . . 4 ((𝑎 = 𝑧𝑏 = 𝑤) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
20 fveq2 6890 . . . . . . 7 (𝑎 = 𝑤 → (𝑇𝑎) = (𝑇𝑤))
21 fveq2 6890 . . . . . . 7 (𝑏 = 𝑧 → (𝑇𝑏) = (𝑇𝑧))
2220, 21eqeqan12d 2740 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑤) = (𝑇𝑧)))
23 eqcom 2733 . . . . . 6 ((𝑇𝑤) = (𝑇𝑧) ↔ (𝑇𝑧) = (𝑇𝑤))
2422, 23bitrdi 286 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
25 eqeq12 2743 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑤 = 𝑧))
26 eqcom 2733 . . . . . 6 (𝑤 = 𝑧𝑧 = 𝑤)
2725, 26bitrdi 286 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑧 = 𝑤))
2824, 27imbi12d 343 . . . 4 ((𝑎 = 𝑤𝑏 = 𝑧) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
29 elfzelz 13546 . . . . . . 7 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3029zred 12709 . . . . . 6 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
3130ssriv 3982 . . . . 5 (1...𝑁) ⊆ ℝ
3231a1i 11 . . . 4 (𝜑 → (1...𝑁) ⊆ ℝ)
33 biidd 261 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → (((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
34 simpr1 1191 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧 ∈ (1...𝑁))
35 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐼𝑛) = (𝐼𝑧))
36 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐽𝑛) = (𝐽𝑧))
3735, 36opeq12d 4879 . . . . . . . . 9 (𝑛 = 𝑧 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
38 opex 5460 . . . . . . . . 9 ⟨(𝐼𝑧), (𝐽𝑧)⟩ ∈ V
3937, 13, 38fvmpt 6998 . . . . . . . 8 (𝑧 ∈ (1...𝑁) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
4034, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
41 simpr2 1192 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ (1...𝑁))
42 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐼𝑛) = (𝐼𝑤))
43 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐽𝑛) = (𝐽𝑤))
4442, 43opeq12d 4879 . . . . . . . . 9 (𝑛 = 𝑤 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
45 opex 5460 . . . . . . . . 9 ⟨(𝐼𝑤), (𝐽𝑤)⟩ ∈ V
4644, 13, 45fvmpt 6998 . . . . . . . 8 (𝑤 ∈ (1...𝑁) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4741, 46syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4840, 47eqeq12d 2742 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) ↔ ⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩))
49 fvex 6903 . . . . . . . 8 (𝐼𝑧) ∈ V
50 fvex 6903 . . . . . . . 8 (𝐽𝑧) ∈ V
5149, 50opth 5472 . . . . . . 7 (⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩ ↔ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)))
5234, 30syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧 ∈ ℝ)
5331, 41sselid 3976 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ ℝ)
54 simpr3 1193 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧𝑤)
5552, 53, 54leltned 11405 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤𝑤𝑧))
562adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝐹:(1...𝑁)–1-1→ℝ)
57 f1fveq 7266 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)–1-1→ℝ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5856, 34, 41, 57syl12anc 835 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5958, 26bitr4di 288 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑤 = 𝑧))
6059necon3bid 2975 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ 𝑤𝑧))
6155, 60bitr4d 281 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 ↔ (𝐹𝑧) ≠ (𝐹𝑤)))
6261biimpa 475 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ≠ (𝐹𝑤))
63 f1f 6787 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (1...𝑁))
6765, 66ffvelcdmd 7088 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ∈ ℝ)
6841adantr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (1...𝑁))
6965, 68ffvelcdmd 7088 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑤) ∈ ℝ)
7067, 69lttri2d 11391 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐹𝑧) ≠ (𝐹𝑤) ↔ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧))))
7162, 70mpbid 231 . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
7572, 73, 3, 4, 66, 68, 74erdszelem8 35036 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐼𝑧) = (𝐼𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7672, 73, 7, 8, 66, 68, 74erdszelem8 35036 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐽𝑧) = (𝐽𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7775, 76anim12d 607 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤))))
78 ioran 981 . . . . . . . . . . . . 13 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
79 fvex 6903 . . . . . . . . . . . . . . . 16 (𝐹𝑧) ∈ V
80 fvex 6903 . . . . . . . . . . . . . . . 16 (𝐹𝑤) ∈ V
8179, 80brcnv 5879 . . . . . . . . . . . . . . 15 ((𝐹𝑧) < (𝐹𝑤) ↔ (𝐹𝑤) < (𝐹𝑧))
8281notbii 319 . . . . . . . . . . . . . 14 (¬ (𝐹𝑧) < (𝐹𝑤) ↔ ¬ (𝐹𝑤) < (𝐹𝑧))
8382anbi2i 621 . . . . . . . . . . . . 13 ((¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
8478, 83bitr4i 277 . . . . . . . . . . . 12 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑧) < (𝐹𝑤)))
8577, 84imbitrrdi 251 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → ¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧))))
8671, 85mt2d 136 . . . . . . . . . 10 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)))
8786ex 411 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8855, 87sylbird 259 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑤𝑧 → ¬ ((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤))))
8988necon4ad 2949 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (((𝐼𝑧) = (𝐼𝑤) ∧ (𝐽𝑧) = (𝐽𝑤)) → 𝑤 = 𝑧))
9051, 89biimtrid 241 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩ → 𝑤 = 𝑧))
9148, 90sylbid 239 . . . . 5 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑤 = 𝑧))
9291, 26imbitrdi 250 . . . 4 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9319, 28, 32, 33, 92wlogle 11785 . . 3 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9493ralrimivva 3191 . 2 (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
95 dff13 7259 . 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 581 1 (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  {crab 3419  wss 3946  𝒫 cpw 4597  cop 4629   class class class wbr 5143  cmpt 5226   × cxp 5670  ccnv 5671  cres 5674  cima 5675  wf 6539  1-1wf1 6540  cfv 6543   Isom wiso 6544  (class class class)co 7413  supcsup 9473  cr 11145  1c1 11147   < clt 11286  cle 11287  cn 12255  ...cfz 13529  chash 14339
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-sup 9475  df-dju 9934  df-card 9972  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12256  df-n0 12516  df-xnn0 12588  df-z 12602  df-uz 12866  df-fz 13530  df-hash 14340
This theorem is referenced by:  erdszelem10  35038
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