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Theorem cdleme31fv 38016
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 7125 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))) ∈ V
31, 2eqeltri 2829 . . 3 𝐶 ∈ V
4 ifexg 4460 . . 3 ((𝐶 ∈ V ∧ 𝑋𝐵) → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
53, 4mpan 690 . 2 (𝑋𝐵 → if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V)
6 breq1 5030 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
76notbid 321 . . . . 5 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
87anbi2d 632 . . . 4 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
9 oveq1 7171 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
109oveq2d 7180 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑠 (𝑥 𝑊)) = (𝑠 (𝑋 𝑊)))
11 id 22 . . . . . . . . . 10 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2754 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑠 (𝑥 𝑊)) = 𝑥 ↔ (𝑠 (𝑋 𝑊)) = 𝑋))
1312anbi2d 632 . . . . . . . 8 (𝑥 = 𝑋 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋)))
149oveq2d 7180 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑁 (𝑥 𝑊)) = (𝑁 (𝑋 𝑊)))
1514eqeq2d 2749 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧 = (𝑁 (𝑥 𝑊)) ↔ 𝑧 = (𝑁 (𝑋 𝑊))))
1613, 15imbi12d 348 . . . . . . 7 (𝑥 = 𝑋 → (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))) ↔ ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
1716ralbidv 3109 . . . . . 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 7123 . . . . 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 2798 . . . 4 (𝑥 = 𝑋𝑂 = 𝐶)
218, 20, 11ifbieq12d 4439 . . 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 6767 . 2 ((𝑋𝐵 ∧ if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋) ∈ V) → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
245, 23mpdan 687 1 (𝑋𝐵 → (𝐹𝑋) = if((𝑃𝑄 ∧ ¬ 𝑋 𝑊), 𝐶, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2113  wne 2934  wral 3053  Vcvv 3397  ifcif 4411   class class class wbr 5027  cmpt 5107  cfv 6333  crio 7120  (class class class)co 7164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-riota 7121  df-ov 7167
This theorem is referenced by:  cdleme31fv1  38017
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