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Theorem erdszelem9 34488
Description: Lemma for erdsze 34491. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n (πœ‘ β†’ 𝑁 ∈ β„•)
erdsze.f (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i 𝐼 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
erdszelem.j 𝐽 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , β—‘ < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
erdszelem.t 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩)
Assertion
Ref Expression
erdszelem9 (πœ‘ β†’ 𝑇:(1...𝑁)–1-1β†’(β„• Γ— β„•))
Distinct variable groups:   π‘₯,𝑦,𝑛,𝐹   𝑛,𝐼,π‘₯,𝑦   𝑛,𝐽,π‘₯,𝑦   𝑛,𝑁,π‘₯,𝑦   πœ‘,𝑛,π‘₯,𝑦
Allowed substitution hints:   𝑇(π‘₯,𝑦,𝑛)
Proof of Theorem erdszelem9
Dummy variables 𝑀 𝑧 π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erdsze.n . . . . . 6 (πœ‘ β†’ 𝑁 ∈ β„•)
2 erdsze.f . . . . . 6 (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)
3 erdszelem.i . . . . . 6 𝐼 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
4 ltso 11298 . . . . . 6 < Or ℝ
51, 2, 3, 4erdszelem6 34485 . . . . 5 (πœ‘ β†’ 𝐼:(1...𝑁)βŸΆβ„•)
65ffvelcdmda 7085 . . . 4 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ (πΌβ€˜π‘›) ∈ β„•)
7 erdszelem.j . . . . . 6 𝐽 = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , β—‘ < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))
8 gtso 11299 . . . . . 6 β—‘ < Or ℝ
91, 2, 7, 8erdszelem6 34485 . . . . 5 (πœ‘ β†’ 𝐽:(1...𝑁)βŸΆβ„•)
109ffvelcdmda 7085 . . . 4 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ (π½β€˜π‘›) ∈ β„•)
11 opelxpi 5712 . . . 4 (((πΌβ€˜π‘›) ∈ β„• ∧ (π½β€˜π‘›) ∈ β„•) β†’ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩ ∈ (β„• Γ— β„•))
126, 10, 11syl2anc 582 . . 3 ((πœ‘ ∧ 𝑛 ∈ (1...𝑁)) β†’ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩ ∈ (β„• Γ— β„•))
13 erdszelem.t . . 3 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩)
1412, 13fmptd 7114 . 2 (πœ‘ β†’ 𝑇:(1...𝑁)⟢(β„• Γ— β„•))
15 fveq2 6890 . . . . . 6 (π‘Ž = 𝑧 β†’ (π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘§))
16 fveq2 6890 . . . . . 6 (𝑏 = 𝑀 β†’ (π‘‡β€˜π‘) = (π‘‡β€˜π‘€))
1715, 16eqeqan12d 2744 . . . . 5 ((π‘Ž = 𝑧 ∧ 𝑏 = 𝑀) β†’ ((π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘) ↔ (π‘‡β€˜π‘§) = (π‘‡β€˜π‘€)))
18 eqeq12 2747 . . . . 5 ((π‘Ž = 𝑧 ∧ 𝑏 = 𝑀) β†’ (π‘Ž = 𝑏 ↔ 𝑧 = 𝑀))
1917, 18imbi12d 343 . . . 4 ((π‘Ž = 𝑧 ∧ 𝑏 = 𝑀) β†’ (((π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘) β†’ π‘Ž = 𝑏) ↔ ((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀)))
20 fveq2 6890 . . . . . . 7 (π‘Ž = 𝑀 β†’ (π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘€))
21 fveq2 6890 . . . . . . 7 (𝑏 = 𝑧 β†’ (π‘‡β€˜π‘) = (π‘‡β€˜π‘§))
2220, 21eqeqan12d 2744 . . . . . 6 ((π‘Ž = 𝑀 ∧ 𝑏 = 𝑧) β†’ ((π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘) ↔ (π‘‡β€˜π‘€) = (π‘‡β€˜π‘§)))
23 eqcom 2737 . . . . . 6 ((π‘‡β€˜π‘€) = (π‘‡β€˜π‘§) ↔ (π‘‡β€˜π‘§) = (π‘‡β€˜π‘€))
2422, 23bitrdi 286 . . . . 5 ((π‘Ž = 𝑀 ∧ 𝑏 = 𝑧) β†’ ((π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘) ↔ (π‘‡β€˜π‘§) = (π‘‡β€˜π‘€)))
25 eqeq12 2747 . . . . . 6 ((π‘Ž = 𝑀 ∧ 𝑏 = 𝑧) β†’ (π‘Ž = 𝑏 ↔ 𝑀 = 𝑧))
26 eqcom 2737 . . . . . 6 (𝑀 = 𝑧 ↔ 𝑧 = 𝑀)
2725, 26bitrdi 286 . . . . 5 ((π‘Ž = 𝑀 ∧ 𝑏 = 𝑧) β†’ (π‘Ž = 𝑏 ↔ 𝑧 = 𝑀))
2824, 27imbi12d 343 . . . 4 ((π‘Ž = 𝑀 ∧ 𝑏 = 𝑧) β†’ (((π‘‡β€˜π‘Ž) = (π‘‡β€˜π‘) β†’ π‘Ž = 𝑏) ↔ ((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀)))
29 elfzelz 13505 . . . . . . 7 (𝑧 ∈ (1...𝑁) β†’ 𝑧 ∈ β„€)
3029zred 12670 . . . . . 6 (𝑧 ∈ (1...𝑁) β†’ 𝑧 ∈ ℝ)
3130ssriv 3985 . . . . 5 (1...𝑁) βŠ† ℝ
3231a1i 11 . . . 4 (πœ‘ β†’ (1...𝑁) βŠ† ℝ)
33 biidd 261 . . . 4 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁))) β†’ (((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀) ↔ ((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀)))
34 simpr1 1192 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ 𝑧 ∈ (1...𝑁))
35 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑧 β†’ (πΌβ€˜π‘›) = (πΌβ€˜π‘§))
36 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑧 β†’ (π½β€˜π‘›) = (π½β€˜π‘§))
3735, 36opeq12d 4880 . . . . . . . . 9 (𝑛 = 𝑧 β†’ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩ = ⟨(πΌβ€˜π‘§), (π½β€˜π‘§)⟩)
38 opex 5463 . . . . . . . . 9 ⟨(πΌβ€˜π‘§), (π½β€˜π‘§)⟩ ∈ V
3937, 13, 38fvmpt 6997 . . . . . . . 8 (𝑧 ∈ (1...𝑁) β†’ (π‘‡β€˜π‘§) = ⟨(πΌβ€˜π‘§), (π½β€˜π‘§)⟩)
4034, 39syl 17 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (π‘‡β€˜π‘§) = ⟨(πΌβ€˜π‘§), (π½β€˜π‘§)⟩)
41 simpr2 1193 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ 𝑀 ∈ (1...𝑁))
42 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑀 β†’ (πΌβ€˜π‘›) = (πΌβ€˜π‘€))
43 fveq2 6890 . . . . . . . . . 10 (𝑛 = 𝑀 β†’ (π½β€˜π‘›) = (π½β€˜π‘€))
4442, 43opeq12d 4880 . . . . . . . . 9 (𝑛 = 𝑀 β†’ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩ = ⟨(πΌβ€˜π‘€), (π½β€˜π‘€)⟩)
45 opex 5463 . . . . . . . . 9 ⟨(πΌβ€˜π‘€), (π½β€˜π‘€)⟩ ∈ V
4644, 13, 45fvmpt 6997 . . . . . . . 8 (𝑀 ∈ (1...𝑁) β†’ (π‘‡β€˜π‘€) = ⟨(πΌβ€˜π‘€), (π½β€˜π‘€)⟩)
4741, 46syl 17 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (π‘‡β€˜π‘€) = ⟨(πΌβ€˜π‘€), (π½β€˜π‘€)⟩)
4840, 47eqeq12d 2746 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ ((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) ↔ ⟨(πΌβ€˜π‘§), (π½β€˜π‘§)⟩ = ⟨(πΌβ€˜π‘€), (π½β€˜π‘€)⟩))
49 fvex 6903 . . . . . . . 8 (πΌβ€˜π‘§) ∈ V
50 fvex 6903 . . . . . . . 8 (π½β€˜π‘§) ∈ V
5149, 50opth 5475 . . . . . . 7 (⟨(πΌβ€˜π‘§), (π½β€˜π‘§)⟩ = ⟨(πΌβ€˜π‘€), (π½β€˜π‘€)⟩ ↔ ((πΌβ€˜π‘§) = (πΌβ€˜π‘€) ∧ (π½β€˜π‘§) = (π½β€˜π‘€)))
5234, 30syl 17 . . . . . . . . . 10 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ 𝑧 ∈ ℝ)
5331, 41sselid 3979 . . . . . . . . . 10 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ 𝑀 ∈ ℝ)
54 simpr3 1194 . . . . . . . . . 10 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ 𝑧 ≀ 𝑀)
5552, 53, 54leltned 11371 . . . . . . . . 9 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (𝑧 < 𝑀 ↔ 𝑀 β‰  𝑧))
562adantr 479 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ 𝐹:(1...𝑁)–1-1→ℝ)
57 f1fveq 7263 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)–1-1→ℝ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁))) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ 𝑧 = 𝑀))
5856, 34, 41, 57syl12anc 833 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ 𝑧 = 𝑀))
5958, 26bitr4di 288 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ 𝑀 = 𝑧))
6059necon3bid 2983 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ ((πΉβ€˜π‘§) β‰  (πΉβ€˜π‘€) ↔ 𝑀 β‰  𝑧))
6155, 60bitr4d 281 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (𝑧 < 𝑀 ↔ (πΉβ€˜π‘§) β‰  (πΉβ€˜π‘€)))
6261biimpa 475 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ (πΉβ€˜π‘§) β‰  (πΉβ€˜π‘€))
63 f1f 6786 . . . . . . . . . . . . . . . 16 (𝐹:(1...𝑁)–1-1→ℝ β†’ 𝐹:(1...𝑁)βŸΆβ„)
642, 63syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐹:(1...𝑁)βŸΆβ„)
6564ad2antrr 722 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ 𝐹:(1...𝑁)βŸΆβ„)
6634adantr 479 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ 𝑧 ∈ (1...𝑁))
6765, 66ffvelcdmd 7086 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ (πΉβ€˜π‘§) ∈ ℝ)
6841adantr 479 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ 𝑀 ∈ (1...𝑁))
6965, 68ffvelcdmd 7086 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ (πΉβ€˜π‘€) ∈ ℝ)
7067, 69lttri2d 11357 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ ((πΉβ€˜π‘§) β‰  (πΉβ€˜π‘€) ↔ ((πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∨ (πΉβ€˜π‘€) < (πΉβ€˜π‘§))))
7162, 70mpbid 231 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ ((πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∨ (πΉβ€˜π‘€) < (πΉβ€˜π‘§)))
721ad2antrr 722 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ 𝑁 ∈ β„•)
732ad2antrr 722 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ 𝐹:(1...𝑁)–1-1→ℝ)
74 simpr 483 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ 𝑧 < 𝑀)
7572, 73, 3, 4, 66, 68, 74erdszelem8 34487 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ ((πΌβ€˜π‘§) = (πΌβ€˜π‘€) β†’ Β¬ (πΉβ€˜π‘§) < (πΉβ€˜π‘€)))
7672, 73, 7, 8, 66, 68, 74erdszelem8 34487 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ ((π½β€˜π‘§) = (π½β€˜π‘€) β†’ Β¬ (πΉβ€˜π‘§)β—‘ < (πΉβ€˜π‘€)))
7775, 76anim12d 607 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ (((πΌβ€˜π‘§) = (πΌβ€˜π‘€) ∧ (π½β€˜π‘§) = (π½β€˜π‘€)) β†’ (Β¬ (πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∧ Β¬ (πΉβ€˜π‘§)β—‘ < (πΉβ€˜π‘€))))
78 ioran 980 . . . . . . . . . . . . 13 (Β¬ ((πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∨ (πΉβ€˜π‘€) < (πΉβ€˜π‘§)) ↔ (Β¬ (πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∧ Β¬ (πΉβ€˜π‘€) < (πΉβ€˜π‘§)))
79 fvex 6903 . . . . . . . . . . . . . . . 16 (πΉβ€˜π‘§) ∈ V
80 fvex 6903 . . . . . . . . . . . . . . . 16 (πΉβ€˜π‘€) ∈ V
8179, 80brcnv 5881 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘§)β—‘ < (πΉβ€˜π‘€) ↔ (πΉβ€˜π‘€) < (πΉβ€˜π‘§))
8281notbii 319 . . . . . . . . . . . . . 14 (Β¬ (πΉβ€˜π‘§)β—‘ < (πΉβ€˜π‘€) ↔ Β¬ (πΉβ€˜π‘€) < (πΉβ€˜π‘§))
8382anbi2i 621 . . . . . . . . . . . . 13 ((Β¬ (πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∧ Β¬ (πΉβ€˜π‘§)β—‘ < (πΉβ€˜π‘€)) ↔ (Β¬ (πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∧ Β¬ (πΉβ€˜π‘€) < (πΉβ€˜π‘§)))
8478, 83bitr4i 277 . . . . . . . . . . . 12 (Β¬ ((πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∨ (πΉβ€˜π‘€) < (πΉβ€˜π‘§)) ↔ (Β¬ (πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∧ Β¬ (πΉβ€˜π‘§)β—‘ < (πΉβ€˜π‘€)))
8577, 84imbitrrdi 251 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ (((πΌβ€˜π‘§) = (πΌβ€˜π‘€) ∧ (π½β€˜π‘§) = (π½β€˜π‘€)) β†’ Β¬ ((πΉβ€˜π‘§) < (πΉβ€˜π‘€) ∨ (πΉβ€˜π‘€) < (πΉβ€˜π‘§))))
8671, 85mt2d 136 . . . . . . . . . 10 (((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) ∧ 𝑧 < 𝑀) β†’ Β¬ ((πΌβ€˜π‘§) = (πΌβ€˜π‘€) ∧ (π½β€˜π‘§) = (π½β€˜π‘€)))
8786ex 411 . . . . . . . . 9 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (𝑧 < 𝑀 β†’ Β¬ ((πΌβ€˜π‘§) = (πΌβ€˜π‘€) ∧ (π½β€˜π‘§) = (π½β€˜π‘€))))
8855, 87sylbird 259 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (𝑀 β‰  𝑧 β†’ Β¬ ((πΌβ€˜π‘§) = (πΌβ€˜π‘€) ∧ (π½β€˜π‘§) = (π½β€˜π‘€))))
8988necon4ad 2957 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (((πΌβ€˜π‘§) = (πΌβ€˜π‘€) ∧ (π½β€˜π‘§) = (π½β€˜π‘€)) β†’ 𝑀 = 𝑧))
9051, 89biimtrid 241 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ (⟨(πΌβ€˜π‘§), (π½β€˜π‘§)⟩ = ⟨(πΌβ€˜π‘€), (π½β€˜π‘€)⟩ β†’ 𝑀 = 𝑧))
9148, 90sylbid 239 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ ((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑀 = 𝑧))
9291, 26imbitrdi 250 . . . 4 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ 𝑧 ≀ 𝑀)) β†’ ((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀))
9319, 28, 32, 33, 92wlogle 11751 . . 3 ((πœ‘ ∧ (𝑧 ∈ (1...𝑁) ∧ 𝑀 ∈ (1...𝑁))) β†’ ((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀))
9493ralrimivva 3198 . 2 (πœ‘ β†’ βˆ€π‘§ ∈ (1...𝑁)βˆ€π‘€ ∈ (1...𝑁)((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀))
95 dff13 7256 . 2 (𝑇:(1...𝑁)–1-1β†’(β„• Γ— β„•) ↔ (𝑇:(1...𝑁)⟢(β„• Γ— β„•) ∧ βˆ€π‘§ ∈ (1...𝑁)βˆ€π‘€ ∈ (1...𝑁)((π‘‡β€˜π‘§) = (π‘‡β€˜π‘€) β†’ 𝑧 = 𝑀)))
9614, 94, 95sylanbrc 581 1 (πœ‘ β†’ 𝑇:(1...𝑁)–1-1β†’(β„• Γ— β„•))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  {crab 3430   βŠ† wss 3947  π’« cpw 4601  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230   Γ— cxp 5673  β—‘ccnv 5674   β†Ύ cres 5677   β€œ cima 5678  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542   Isom wiso 6543  (class class class)co 7411  supcsup 9437  β„cr 11111  1c1 11113   < clt 11252   ≀ cle 11253  β„•cn 12216  ...cfz 13488  β™―chash 14294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-hash 14295
This theorem is referenced by:  erdszelem10  34489
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