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Theorem erdszelem1 35196
Description: Lemma for erdsze 35207. (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 7464 . . . 4 (1...𝐴) ∈ V
21elpw2 5334 . . 3 (𝑋 ∈ 𝒫 (1...𝐴) ↔ 𝑋 ⊆ (1...𝐴))
32anbi1i 624 . 2 ((𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)) ↔ (𝑋 ⊆ (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
4 reseq2 5992 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
5 isoeq1 7337 . . . . . 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 7340 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑋) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑦))))
8 imaeq2 6074 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
9 isoeq5 7341 . . . . . 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 305 . . . 4 (𝑦 = 𝑋 → ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋))))
12 eleq2 2830 . . . 4 (𝑦 = 𝑋 → (𝐴𝑦𝐴𝑋))
1311, 12anbi12d 632 . . 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 3695 . 2 (𝑋𝑆 ↔ (𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
16 3anass 1095 . 2 ((𝑋 ⊆ (1...𝐴) ∧ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋) ↔ (𝑋 ⊆ (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
173, 15, 163bitr4i 303 1 (𝑋𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  wss 3951  𝒫 cpw 4600  cres 5687  cima 5688   Isom wiso 6562  (class class class)co 7431  1c1 11156   < clt 11295  ...cfz 13547
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-ov 7434
This theorem is referenced by:  erdszelem2  35197  erdszelem4  35199  erdszelem7  35202  erdszelem8  35203
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