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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme31id | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 18-Apr-2013.) |
Ref | Expression |
---|---|
cdleme31fv2.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥)) |
Ref | Expression |
---|---|
cdleme31id | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . 3 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊) → 𝑃 ≠ 𝑄) | |
2 | 1 | necon2bi 2961 | . 2 ⊢ (𝑃 = 𝑄 → ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) |
3 | cdleme31fv2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), 𝑂, 𝑥)) | |
4 | 3 | cdleme31fv2 39922 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ (𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) |
5 | 2, 4 | sylan2 591 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑃 = 𝑄) → (𝐹‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ifcif 4524 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: cdleme32fvaw 39968 cdleme42keg 40015 cdleme42mgN 40017 cdleme17d4 40026 cdleme48fvg 40029 cdleme50trn3 40082 cdlemg1idlemN 40101 cdlemg2idN 40125 |
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