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 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
cdleme31.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
cdleme31.c	⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))

Assertion

Ref	Expression
cdleme31fv	⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = if((𝑃 ≠ 𝑄 ∧ ¬ 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

Step	Hyp	Ref	Expression
1		cdleme31.c	. . . 4 ⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))
2		riotaex 7125	. . . 4 ⊢ (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))) ∈ V
3	1, 2	eqeltri 2829	. . 3 ⊢ 𝐶 ∈ V
4		ifexg 4460	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝑋 ∈ 𝐵) → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋) ∈ V)
5	3, 4	mpan 690	. 2 ⊢ (𝑋 ∈ 𝐵 → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋) ∈ V)
6		breq1 5030	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
7	6	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ≤ 𝑊 ↔ ¬ 𝑋 ≤ 𝑊))
8	7	anbi2d 632	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)))
9		oveq1 7171	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑊) = (𝑋 ∧ 𝑊))
10	9	oveq2d 7180	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑠 ∨ (𝑥 ∧ 𝑊)) = (𝑠 ∨ (𝑋 ∧ 𝑊)))
11		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2754	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥 ↔ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
13	12	anbi2d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
14	9	oveq2d 7180	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑁 ∨ (𝑥 ∧ 𝑊)) = (𝑁 ∨ (𝑋 ∧ 𝑊)))
15	14	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)) ↔ 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))
16	13, 15	imbi12d 348	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))) ↔ ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
17	16	ralbidv 3109	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))) ↔ ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
18	17	riotabidv 7123	. . . . 5 ⊢ (𝑥 = 𝑋 → (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
19		cdleme31.o	. . . . 5 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
20	18, 19, 1	3eqtr4g 2798	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑂 = 𝐶)
21	8, 20, 11	ifbieq12d 4439	. . 3 ⊢ (𝑥 = 𝑋 → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥) = if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋))
22		cdleme31.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
23	21, 22	fvmptg 6767	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋) ∈ V) → (𝐹‘𝑋) = if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋))
24	5, 23	mpdan 687	1 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋))

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