Theorem cdleme31fv 41222

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 7380	. . . 4 ⊢ (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))) ∈ V
3	1, 2	eqeltri 2861	. . 3 ⊢ 𝐶 ∈ V
4		ifexg 4539	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝑋 ∈ 𝐵) → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋) ∈ V)
5	3, 4	mpan 703	. 2 ⊢ (𝑋 ∈ 𝐵 → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋) ∈ V)
6		breq1 5114	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
7	6	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ≤ 𝑊 ↔ ¬ 𝑋 ≤ 𝑊))
8	7	anbi2d 642	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)))
9		oveq1 7426	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑊) = (𝑋 ∧ 𝑊))
10	9	oveq2d 7435	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑠 ∨ (𝑥 ∧ 𝑊)) = (𝑠 ∨ (𝑋 ∧ 𝑊)))
11		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥 ↔ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
13	12	anbi2d 642	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) ↔ (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
14	9	oveq2d 7435	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑁 ∨ (𝑥 ∧ 𝑊)) = (𝑁 ∨ (𝑋 ∧ 𝑊)))
15	14	eqeq2d 2776	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)) ↔ 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))
16	13, 15	imbi12d 347	. . . . . . 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 3190	. . . . . 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 7378	. . . . 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 2825	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑂 = 𝐶)
21	8, 20, 11	ifbieq12d 4518	. . 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 6991	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋) ∈ V) → (𝐹‘𝑋) = if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋))
24	5, 23	mpdan 700	1 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  Vcvv 3457  ifcif 4489   class class class wbr 5111   ↦ cmpt 5194  ‘cfv 6540  ℩crio 7375  (class class class)co 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-riota 7376  df-ov 7422

This theorem is used by:  cdleme31fv1  41223