Theorem cdleme31fv2 40690

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 5102	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
3	2	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (¬ 𝑥 ≤ 𝑊 ↔ ¬ 𝑋 ≤ 𝑊))
4	3	anbi2d 631	. . . . . . 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 715	. . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑥 = 𝑋) → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊))
8	7	iffalsed 4491	. . 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 2772	. 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 6951	1 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ifcif 4480   class class class wbr 5099   ↦ cmpt 5180  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501

This theorem is referenced by:  cdleme31id  40691  cdleme32fvcl  40737  cdleme32e  40742  cdleme32le  40744  cdleme48gfv  40834  cdleme50ldil  40845