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Theorem cdleme31fv 35497
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 6600 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))) ∈ V
31, 2eqeltri 2695 . . 3 𝐶 ∈ V
4 ifexg 4148 . . 3 ((𝐶 ∈ V ∧ 𝑋𝐵) → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
53, 4mpan 705 . 2 (𝑋𝐵 → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
6 breq1 4647 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
76notbid 308 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
87anbi2d 739 . . . 4 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
9 oveq1 6642 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
109oveq2d 6651 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑠 (𝑥 𝑊)) = (𝑠 (𝑋 𝑊)))
11 id 22 . . . . . . . . . 10 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2635 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑠 (𝑥 𝑊)) = 𝑥 ↔ (𝑠 (𝑋 𝑊)) = 𝑋))
1312anbi2d 739 . . . . . . . 8 (𝑥 = 𝑋 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋)))
149oveq2d 6651 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑁 (𝑥 𝑊)) = (𝑁 (𝑋 𝑊)))
1514eqeq2d 2630 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧 = (𝑁 (𝑥 𝑊)) ↔ 𝑧 = (𝑁 (𝑋 𝑊))))
1613, 15imbi12d 334 . . . . . . 7 (𝑥 = 𝑋 → (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))) ↔ ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
1716ralbidv 2983 . . . . . 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 6598 . . . . 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 2679 . . . 4 (𝑥 = 𝑋𝑂 = 𝐶)
218, 20, 11ifbieq12d 4104 . . 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 6267 . 2 ((𝑋𝐵 ∧ if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V) → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
245, 23mpdan 701 1 (𝑋𝐵 → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  Vcvv 3195  ifcif 4077   class class class wbr 4644  cmpt 4720  cfv 5876  crio 6595  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-riota 6596  df-ov 6638
This theorem is referenced by:  cdleme31fv1  35498
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