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Theorem erdszelem1 31147
Description: Lemma for erdsze 31158. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypothesis
Ref Expression
erdszelem1.1 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}
Assertion
Ref Expression
erdszelem1 (𝑋𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑂   𝑦,𝑋
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem erdszelem1
StepHypRef Expression
1 ovex 6663 . . . 4 (1...𝐴) ∈ V
21elpw2 4819 . . 3 (𝑋 ∈ 𝒫 (1...𝐴) ↔ 𝑋 ⊆ (1...𝐴))
32anbi1i 730 . 2 ((𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)) ↔ (𝑋 ⊆ (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
4 reseq2 5380 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
5 isoeq1 6552 . . . . . 6 ((𝐹𝑦) = (𝐹𝑋) → ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑦, (𝐹𝑦))))
64, 5syl 17 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑦, (𝐹𝑦))))
7 isoeq4 6555 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑋) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑦))))
8 imaeq2 5450 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
9 isoeq5 6556 . . . . . 6 ((𝐹𝑦) = (𝐹𝑋) → ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋))))
108, 9syl 17 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋))))
116, 7, 103bitrd 294 . . . 4 (𝑦 = 𝑋 → ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋))))
12 eleq2 2688 . . . 4 (𝑦 = 𝑋 → (𝐴𝑦𝐴𝑋))
1311, 12anbi12d 746 . . 3 (𝑦 = 𝑋 → (((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦) ↔ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
14 erdszelem1.1 . . 3 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}
1513, 14elrab2 3360 . 2 (𝑋𝑆 ↔ (𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
16 3anass 1040 . 2 ((𝑋 ⊆ (1...𝐴) ∧ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋) ↔ (𝑋 ⊆ (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
173, 15, 163bitr4i 292 1 (𝑋𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  {crab 2913  wss 3567  𝒫 cpw 4149  cres 5106  cima 5107   Isom wiso 5877  (class class class)co 6635  1c1 9922   < clt 10059  ...cfz 12311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-ov 6638
This theorem is referenced by:  erdszelem2  31148  erdszelem4  31150  erdszelem7  31153  erdszelem8  31154
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