Theorem cdleme31fv2 41200

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 23-Feb-2013.)

Hypothesis

Ref	Expression
cdleme31fv2.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))

Assertion

Ref	Expression
cdleme31fv2	⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Distinct variable groups:   𝑥,𝐵   𝑥, ≤   𝑥,𝑃   𝑥,𝑄   𝑥,𝑊   𝑥,𝑋

Allowed substitution hints:   𝐹(𝑥)   𝑂(𝑥)

Proof of Theorem cdleme31fv2

Step	Hyp	Ref	Expression
1		cdleme31fv2.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
2		breq1 5114	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
3	2	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ≤ 𝑊 ↔ ¬ 𝑋 ≤ 𝑊))
4	3	anbi2d 642	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)))
5	4	notbid 321	. . . . . 6 ⊢ (𝑥 = 𝑋 → (¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊) ↔ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)))
6	5	biimparc 485	. . . . 5 ⊢ ((¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑥 = 𝑋) → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊))
7	6	adantll 727	. . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊))
8	7	iffalsed 4500	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥) = 𝑥)
9		simpr 490	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
10	8, 9	eqtrd 2800	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥) = 𝑋)
11		simpl 488	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
12	1, 10, 11, 11	fvmptd2 7002	1 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ifcif 4489   class class class wbr 5111   ↦ cmpt 5194  ‘cfv 6540

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-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-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  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

This theorem is used by:  cdleme31id  41201  cdleme32fvcl  41247  cdleme32e  41252  cdleme32le  41254  cdleme48gfv  41344  cdleme50ldil  41355