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Mirrors > Home > MPE Home > Th. List > resdisj | Structured version Visualization version GIF version |
Description: A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
resdisj | ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseq2 5976 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ↾ (𝐴 ∩ 𝐵)) = (𝐶 ↾ ∅)) | |
2 | resres 5994 | . 2 ⊢ ((𝐶 ↾ 𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴 ∩ 𝐵)) | |
3 | res0 5985 | . . 3 ⊢ (𝐶 ↾ ∅) = ∅ | |
4 | 3 | eqcomi 2741 | . 2 ⊢ ∅ = (𝐶 ↾ ∅) |
5 | 1, 2, 4 | 3eqtr4g 2797 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∩ cin 3947 ∅c0 4322 ↾ cres 5678 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-opab 5211 df-xp 5682 df-rel 5683 df-res 5688 |
This theorem is referenced by: fvsnun1 7179 |
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