Theorem cdleme31fv2 40438

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 5094	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
3	2	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ≤ 𝑊 ↔ ¬ 𝑋 ≤ 𝑊))
4	3	anbi2d 630	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)))
5	4	notbid 318	. . . . . 6 ⊢ (𝑥 = 𝑋 → (¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊) ↔ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)))
6	5	biimparc 479	. . . . 5 ⊢ ((¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑥 = 𝑋) → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊))
7	6	adantll 714	. . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊))
8	7	iffalsed 4486	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥) = 𝑥)
9		simpr 484	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
10	8, 9	eqtrd 2766	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥) = 𝑋)
11		simpl 482	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
12	1, 10, 11, 11	fvmptd2 6937	1 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ifcif 4475   class class class wbr 5091   ↦ cmpt 5172  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by:  cdleme31id  40439  cdleme32fvcl  40485  cdleme32e  40490  cdleme32le  40492  cdleme48gfv  40582  cdleme50ldil  40593