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Theorem cdleme31fv 39564
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)
Hypotheses
Ref Expression
cdleme31.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme31.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
cdleme31.c 𝐶 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊))))
Assertion
Ref Expression
cdleme31fv (𝑋𝐵 → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,   𝑥,𝑃   𝑥,𝑄   𝑥,𝑊   𝑥,𝑠,𝑧,𝑋
Allowed substitution hints:   𝐴(𝑥,𝑧,𝑠)   𝐵(𝑧,𝑠)   𝐶(𝑧,𝑠)   𝑃(𝑧,𝑠)   𝑄(𝑧,𝑠)   𝐹(𝑥,𝑧,𝑠)   (𝑥,𝑧,𝑠)   (𝑧,𝑠)   (𝑥,𝑧,𝑠)   𝑁(𝑥,𝑧,𝑠)   𝑂(𝑥,𝑧,𝑠)   𝑊(𝑧,𝑠)

Proof of Theorem cdleme31fv
StepHypRef Expression
1 cdleme31.c . . . 4 𝐶 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊))))
2 riotaex 7371 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))) ∈ V
31, 2eqeltri 2827 . . 3 𝐶 ∈ V
4 ifexg 4576 . . 3 ((𝐶 ∈ V ∧ 𝑋𝐵) → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
53, 4mpan 686 . 2 (𝑋𝐵 → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
6 breq1 5150 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
76notbid 317 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
87anbi2d 627 . . . 4 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
9 oveq1 7418 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
109oveq2d 7427 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑠 (𝑥 𝑊)) = (𝑠 (𝑋 𝑊)))
11 id 22 . . . . . . . . . 10 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2746 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑠 (𝑥 𝑊)) = 𝑥 ↔ (𝑠 (𝑋 𝑊)) = 𝑋))
1312anbi2d 627 . . . . . . . 8 (𝑥 = 𝑋 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋)))
149oveq2d 7427 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑁 (𝑥 𝑊)) = (𝑁 (𝑋 𝑊)))
1514eqeq2d 2741 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧 = (𝑁 (𝑥 𝑊)) ↔ 𝑧 = (𝑁 (𝑋 𝑊))))
1613, 15imbi12d 343 . . . . . . 7 (𝑥 = 𝑋 → (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))) ↔ ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
1716ralbidv 3175 . . . . . 6 (𝑥 = 𝑋 → (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))) ↔ ∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
1817riotabidv 7369 . . . . 5 (𝑥 = 𝑋 → (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊)))) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
19 cdleme31.o . . . . 5 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
2018, 19, 13eqtr4g 2795 . . . 4 (𝑥 = 𝑋𝑂 = 𝐶)
218, 20, 11ifbieq12d 4555 . . 3 (𝑥 = 𝑋 → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
22 cdleme31.f . . 3 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
2321, 22fvmptg 6995 . 2 ((𝑋𝐵 ∧ if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V) → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
245, 23mpdan 683 1 (𝑋𝐵 → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wcel 2104  wne 2938  wral 3059  Vcvv 3472  ifcif 4527   class class class wbr 5147  cmpt 5230  cfv 6542  crio 7366  (class class class)co 7411
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-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  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-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7367  df-ov 7414
This theorem is referenced by:  cdleme31fv1  39565
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