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Theorem erdszelem9 31307
 Description: Lemma for erdsze 31310. (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 10156 . . . . . 6 < Or ℝ
51, 2, 3, 4erdszelem6 31304 . . . . 5 (𝜑𝐼:(1...𝑁)⟶ℕ)
65ffvelrnda 6399 . . . 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 10157 . . . . . 6 < Or ℝ
91, 2, 7, 8erdszelem6 31304 . . . . 5 (𝜑𝐽:(1...𝑁)⟶ℕ)
109ffvelrnda 6399 . . . 4 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐽𝑛) ∈ ℕ)
11 opelxpi 5182 . . . 4 (((𝐼𝑛) ∈ ℕ ∧ (𝐽𝑛) ∈ ℕ) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
126, 10, 11syl2anc 694 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ⟨(𝐼𝑛), (𝐽𝑛)⟩ ∈ (ℕ × ℕ))
13 erdszelem.t . . 3 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)
1412, 13fmptd 6425 . 2 (𝜑𝑇:(1...𝑁)⟶(ℕ × ℕ))
15 fveq2 6229 . . . . . 6 (𝑎 = 𝑧 → (𝑇𝑎) = (𝑇𝑧))
16 fveq2 6229 . . . . . 6 (𝑏 = 𝑤 → (𝑇𝑏) = (𝑇𝑤))
1715, 16eqeqan12d 2667 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
18 eqeq12 2664 . . . . 5 ((𝑎 = 𝑧𝑏 = 𝑤) → (𝑎 = 𝑏𝑧 = 𝑤))
1917, 18imbi12d 333 . . . 4 ((𝑎 = 𝑧𝑏 = 𝑤) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
20 fveq2 6229 . . . . . . 7 (𝑎 = 𝑤 → (𝑇𝑎) = (𝑇𝑤))
21 fveq2 6229 . . . . . . 7 (𝑏 = 𝑧 → (𝑇𝑏) = (𝑇𝑧))
2220, 21eqeqan12d 2667 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑤) = (𝑇𝑧)))
23 eqcom 2658 . . . . . 6 ((𝑇𝑤) = (𝑇𝑧) ↔ (𝑇𝑧) = (𝑇𝑤))
2422, 23syl6bb 276 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → ((𝑇𝑎) = (𝑇𝑏) ↔ (𝑇𝑧) = (𝑇𝑤)))
25 eqeq12 2664 . . . . . 6 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑤 = 𝑧))
26 eqcom 2658 . . . . . 6 (𝑤 = 𝑧𝑧 = 𝑤)
2725, 26syl6bb 276 . . . . 5 ((𝑎 = 𝑤𝑏 = 𝑧) → (𝑎 = 𝑏𝑧 = 𝑤))
2824, 27imbi12d 333 . . . 4 ((𝑎 = 𝑤𝑏 = 𝑧) → (((𝑇𝑎) = (𝑇𝑏) → 𝑎 = 𝑏) ↔ ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤)))
29 elfzelz 12380 . . . . . . 7 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ)
3029zred 11520 . . . . . 6 (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℝ)
3130ssriv 3640 . . . . 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 1087 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧 ∈ (1...𝑁))
35 fveq2 6229 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐼𝑛) = (𝐼𝑧))
36 fveq2 6229 . . . . . . . . . 10 (𝑛 = 𝑧 → (𝐽𝑛) = (𝐽𝑧))
3735, 36opeq12d 4441 . . . . . . . . 9 (𝑛 = 𝑧 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
38 opex 4962 . . . . . . . . 9 ⟨(𝐼𝑧), (𝐽𝑧)⟩ ∈ V
3937, 13, 38fvmpt 6321 . . . . . . . 8 (𝑧 ∈ (1...𝑁) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
4034, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑧) = ⟨(𝐼𝑧), (𝐽𝑧)⟩)
41 simpr2 1088 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ (1...𝑁))
42 fveq2 6229 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐼𝑛) = (𝐼𝑤))
43 fveq2 6229 . . . . . . . . . 10 (𝑛 = 𝑤 → (𝐽𝑛) = (𝐽𝑤))
4442, 43opeq12d 4441 . . . . . . . . 9 (𝑛 = 𝑤 → ⟨(𝐼𝑛), (𝐽𝑛)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
45 opex 4962 . . . . . . . . 9 ⟨(𝐼𝑤), (𝐽𝑤)⟩ ∈ V
4644, 13, 45fvmpt 6321 . . . . . . . 8 (𝑤 ∈ (1...𝑁) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4741, 46syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑇𝑤) = ⟨(𝐼𝑤), (𝐽𝑤)⟩)
4840, 47eqeq12d 2666 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → ((𝑇𝑧) = (𝑇𝑤) ↔ ⟨(𝐼𝑧), (𝐽𝑧)⟩ = ⟨(𝐼𝑤), (𝐽𝑤)⟩))
49 fvex 6239 . . . . . . . 8 (𝐼𝑧) ∈ V
50 fvex 6239 . . . . . . . 8 (𝐽𝑧) ∈ V
5149, 50opth 4974 . . . . . . 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 3634 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑤 ∈ ℝ)
54 simpr3 1089 . . . . . . . . . 10 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝑧𝑤)
5552, 53, 54leltned 10228 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → (𝑧 < 𝑤𝑤𝑧))
562adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) → 𝐹:(1...𝑁)–1-1→ℝ)
57 f1fveq 6559 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)–1-1→ℝ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5856, 34, 41, 57syl12anc 1364 . . . . . . . . . . . . . . . 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 2867 . . . . . . . . . . . . . 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 500 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ≠ (𝐹𝑤))
63 f1f 6139 . . . . . . . . . . . . . . . 16 (𝐹:(1...𝑁)–1-1→ℝ → 𝐹:(1...𝑁)⟶ℝ)
642, 63syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:(1...𝑁)⟶ℝ)
6564ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)⟶ℝ)
6634adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (1...𝑁))
6765, 66ffvelrnd 6400 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑧) ∈ ℝ)
6841adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (1...𝑁))
6965, 68ffvelrnd 6400 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → (𝐹𝑤) ∈ ℝ)
7067, 69lttri2d 10214 . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑁 ∈ ℕ)
732ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝐹:(1...𝑁)–1-1→ℝ)
74 simpr 476 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
7572, 73, 3, 4, 66, 68, 74erdszelem8 31306 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁) ∧ 𝑧𝑤)) ∧ 𝑧 < 𝑤) → ((𝐼𝑧) = (𝐼𝑤) → ¬ (𝐹𝑧) < (𝐹𝑤)))
7672, 73, 7, 8, 66, 68, 74erdszelem8 31306 . . . . . . . . . . . . 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 510 . . . . . . . . . . . . 13 (¬ ((𝐹𝑧) < (𝐹𝑤) ∨ (𝐹𝑤) < (𝐹𝑧)) ↔ (¬ (𝐹𝑧) < (𝐹𝑤) ∧ ¬ (𝐹𝑤) < (𝐹𝑧)))
79 fvex 6239 . . . . . . . . . . . . . . . 16 (𝐹𝑧) ∈ V
80 fvex 6239 . . . . . . . . . . . . . . . 16 (𝐹𝑤) ∈ V
8179, 80brcnv 5337 . . . . . . . . . . . . . . 15 ((𝐹𝑧) < (𝐹𝑤) ↔ (𝐹𝑤) < (𝐹𝑧))
8281notbii 309 . . . . . . . . . . . . . 14 (¬ (𝐹𝑧) < (𝐹𝑤) ↔ ¬ (𝐹𝑤) < (𝐹𝑧))
8382anbi2i 730 . . . . . . . . . . . . 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 449 . . . . . . . . 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 2842 . . . . . . 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 10599 . . 3 ((𝜑 ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑤 ∈ (1...𝑁))) → ((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
9493ralrimivva 3000 . 2 (𝜑 → ∀𝑧 ∈ (1...𝑁)∀𝑤 ∈ (1...𝑁)((𝑇𝑧) = (𝑇𝑤) → 𝑧 = 𝑤))
95 dff13 6552 . 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 699 1 (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  {crab 2945   ⊆ wss 3607  𝒫 cpw 4191  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  ◡ccnv 5142   ↾ cres 5145   “ cima 5146  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926   Isom wiso 5927  (class class class)co 6690  supcsup 8387  ℝcr 9973  1c1 9975   < clt 10112   ≤ cle 10113  ℕcn 11058  ...cfz 12364  #chash 13157 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158 This theorem is referenced by:  erdszelem10  31308
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