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Theorem cdleme31fv1s 36466
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-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))
Assertion
Ref Expression
cdleme31fv1s ((𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋 / 𝑥𝑂)
Distinct variable groups:   𝑥,𝐵   𝑥,   𝑥,𝑃   𝑥,𝑄   𝑥,𝑊   𝑥,𝑠,𝑧,𝑋   𝑥,𝐴   𝐵,𝑠,𝑧   𝑥,   𝑥,   𝑥,𝑁
Allowed substitution hints:   𝐴(𝑧,𝑠)   𝑃(𝑧,𝑠)   𝑄(𝑧,𝑠)   𝐹(𝑥,𝑧,𝑠)   (𝑧,𝑠)   (𝑧,𝑠)   (𝑧,𝑠)   𝑁(𝑧,𝑠)   𝑂(𝑥,𝑧,𝑠)   𝑊(𝑧,𝑠)

Proof of Theorem cdleme31fv1s
StepHypRef Expression
1 cdleme31.o . . 3 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
2 cdleme31.f . . 3 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
3 eqid 2825 . . 3 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊))))
41, 2, 3cdleme31fv1 36465 . 2 ((𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
51, 3cdleme31so 36453 . . 3 (𝑋𝐵𝑋 / 𝑥𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
65adantr 474 . 2 ((𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋 / 𝑥𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
74, 6eqtr4d 2864 1 ((𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋 / 𝑥𝑂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wcel 2164  wne 2999  wral 3117  csb 3757  ifcif 4308   class class class wbr 4875  cmpt 4954  cfv 6127  crio 6870  (class class class)co 6910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-riota 6871  df-ov 6913
This theorem is referenced by:  cdlemefrs32fva1  36475  cdleme32fva1  36512
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