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Theorem cdleme31fv 34495
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 6489 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))) ∈ V
31, 2eqeltri 2679 . . 3 𝐶 ∈ V
4 ifexg 4102 . . 3 ((𝐶 ∈ V ∧ 𝑋𝐵) → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
53, 4mpan 701 . 2 (𝑋𝐵 → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
6 breq1 4576 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
76notbid 306 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
87anbi2d 735 . . . 4 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
9 oveq1 6530 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
109oveq2d 6539 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑠 (𝑥 𝑊)) = (𝑠 (𝑋 𝑊)))
11 id 22 . . . . . . . . . 10 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2620 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑠 (𝑥 𝑊)) = 𝑥 ↔ (𝑠 (𝑋 𝑊)) = 𝑋))
1312anbi2d 735 . . . . . . . 8 (𝑥 = 𝑋 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋)))
149oveq2d 6539 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑁 (𝑥 𝑊)) = (𝑁 (𝑋 𝑊)))
1514eqeq2d 2615 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧 = (𝑁 (𝑥 𝑊)) ↔ 𝑧 = (𝑁 (𝑋 𝑊))))
1613, 15imbi12d 332 . . . . . . 7 (𝑥 = 𝑋 → (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))) ↔ ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
1716ralbidv 2964 . . . . . 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 6487 . . . . 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 2664 . . . 4 (𝑥 = 𝑋𝑂 = 𝐶)
218, 20, 11ifbieq12d 4058 . . 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 6170 . 2 ((𝑋𝐵 ∧ if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V) → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
245, 23mpdan 698 1 (𝑋𝐵 → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  wne 2775  wral 2891  Vcvv 3168  ifcif 4031   class class class wbr 4573  cmpt 4633  cfv 5786  crio 6484  (class class class)co 6523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-riota 6485  df-ov 6526
This theorem is referenced by:  cdleme31fv1  34496
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