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Theorem cdleme31fv 38882
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 7322 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))) ∈ V
31, 2eqeltri 2834 . . 3 𝐶 ∈ V
4 ifexg 4540 . . 3 ((𝐶 ∈ V ∧ 𝑋𝐵) → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
53, 4mpan 689 . 2 (𝑋𝐵 → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
6 breq1 5113 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
76notbid 318 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
87anbi2d 630 . . . 4 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
9 oveq1 7369 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
109oveq2d 7378 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑠 (𝑥 𝑊)) = (𝑠 (𝑋 𝑊)))
11 id 22 . . . . . . . . . 10 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2753 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑠 (𝑥 𝑊)) = 𝑥 ↔ (𝑠 (𝑋 𝑊)) = 𝑋))
1312anbi2d 630 . . . . . . . 8 (𝑥 = 𝑋 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋)))
149oveq2d 7378 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑁 (𝑥 𝑊)) = (𝑁 (𝑋 𝑊)))
1514eqeq2d 2748 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧 = (𝑁 (𝑥 𝑊)) ↔ 𝑧 = (𝑁 (𝑋 𝑊))))
1613, 15imbi12d 345 . . . . . . 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 7320 . . . . 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 2802 . . . 4 (𝑥 = 𝑋𝑂 = 𝐶)
218, 20, 11ifbieq12d 4519 . . 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 6951 . 2 ((𝑋𝐵 ∧ if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V) → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
245, 23mpdan 686 1 (𝑋𝐵 → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  Vcvv 3448  ifcif 4491   class class class wbr 5110  cmpt 5193  cfv 6501  crio 7317  (class class class)co 7362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-riota 7318  df-ov 7365
This theorem is referenced by:  cdleme31fv1  38883
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