Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erdszelem1 Structured version   Visualization version   GIF version

Theorem erdszelem1 31299
Description: Lemma for erdsze 31310. (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 6718 . . . 4 (1...𝐴) ∈ V
21elpw2 4858 . . 3 (𝑋 ∈ 𝒫 (1...𝐴) ↔ 𝑋 ⊆ (1...𝐴))
32anbi1i 731 . 2 ((𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)) ↔ (𝑋 ⊆ (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
4 reseq2 5423 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
5 isoeq1 6607 . . . . . 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 6610 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑋) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ↔ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑦))))
8 imaeq2 5497 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
9 isoeq5 6611 . . . . . 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 2719 . . . 4 (𝑦 = 𝑋 → (𝐴𝑦𝐴𝑋))
1311, 12anbi12d 747 . . 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 3399 . 2 (𝑋𝑆 ↔ (𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋)))
16 3anass 1059 . 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 383  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  wss 3607  𝒫 cpw 4191  cres 5145  cima 5146   Isom wiso 5927  (class class class)co 6690  1c1 9975   < clt 10112  ...cfz 12364
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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  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-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-ov 6693
This theorem is referenced by:  erdszelem2  31300  erdszelem4  31302  erdszelem7  31305  erdszelem8  31306
  Copyright terms: Public domain W3C validator