Theorem cdleme31fv1 41555

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
cdleme31fv1	⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝐶)

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 cdleme31fv1

Step	Hyp	Ref	Expression
1		cdleme31.o	. . 3 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊))))
2		cdleme31.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
3		cdleme31.c	. . 3 ⊢ 𝐶 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊))))
4	1, 2, 3	cdleme31fv 41554	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋))
5		iftrue 4487	. 2 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊) → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊), 𝐶, 𝑋) = 𝐶)
6	4, 5	sylan9eq 2815	1 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6541  ℩crio 7378  (class class class)co 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6496  df-fun 6543  df-fv 6549  df-riota 7379  df-ov 7425

This theorem is used by:  cdleme31fv1s  41556  cdleme32fvcl  41604  cdleme32a  41605  cdleme42b  41642