Theorem cdleme31id 40593

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 482	. . 3 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊) → 𝑃 ≠ 𝑄)
2	1	necon2bi 2960	. 2 ⊢ (𝑃 = 𝑄 → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊))
3		cdleme31fv2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥))
4	3	cdleme31fv2 40592	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)
5	2, 4	sylan2 593	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ifcif 4477   class class class wbr 5096   ↦ cmpt 5177  ‘cfv 6490

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498

This theorem is referenced by:  cdleme32fvaw  40638  cdleme42keg  40685  cdleme42mgN  40687  cdleme17d4  40696  cdleme48fvg  40699  cdleme50trn3  40752  cdlemg1idlemN  40771  cdlemg2idN  40795