Theorem cdleme31fv1s 41265

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-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))

Assertion

Ref	Expression
cdleme31fv1s	⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = ⦋𝑋 / 𝑥⦌𝑂)

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

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		eqid 2760	. . 3 ⊢ (℩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))))
4	1, 2, 3	cdleme31fv1 41264	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
5	1, 3	cdleme31so 41252	. . 3 ⊢ (𝑋 ∈ 𝐵 → ⦋𝑋 / 𝑥⦌𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
6	5	adantr 486	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → ⦋𝑋 / 𝑥⦌𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑧 = (𝑁 ∨ (𝑋 ∧ 𝑊)))))
7	4, 6	eqtr4d 2798	1 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = ⦋𝑋 / 𝑥⦌𝑂)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ⦋csb 3847  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6533  ℩crio 7369  (class class class)co 7413

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 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-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-riota 7370  df-ov 7416

This theorem is used by:  cdlemefrs32fva1  41274  cdleme32fva1  41311