Theorem cdleme31id 40899

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 18-Apr-2013.)

Hypothesis

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

Assertion

Ref	Expression
cdleme31id	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → (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 cdleme31id

Step	Hyp	Ref	Expression
1		simpl 484	. . 3 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊) → 𝑃 ≠ 𝑄)
2	1	necon2bi 2966	. 2 ⊢ (𝑃 = 𝑄 → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊))
3		cdleme31fv2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
4	3	cdleme31fv2 40898	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)
5	2, 4	sylan2 600	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ifcif 4456   class class class wbr 5074   ↦ cmpt 5155  ‘cfv 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6444  df-fun 6490  df-fv 6496

This theorem is referenced by:  cdleme32fvaw  40944  cdleme42keg  40991  cdleme42mgN  40993  cdleme17d4  41002  cdleme48fvg  41005  cdleme50trn3  41058  cdlemg1idlemN  41077  cdlemg2idN  41101