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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 5813 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ↾ (𝐴 ∩ 𝐵)) = (𝐶 ↾ ∅)) | |
2 | resres 5831 | . 2 ⊢ ((𝐶 ↾ 𝐴) ↾ 𝐵) = (𝐶 ↾ (𝐴 ∩ 𝐵)) | |
3 | res0 5822 | . . 3 ⊢ (𝐶 ↾ ∅) = ∅ | |
4 | 3 | eqcomi 2807 | . 2 ⊢ ∅ = (𝐶 ↾ ∅) |
5 | 1, 2, 4 | 3eqtr4g 2858 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∩ cin 3880 ∅c0 4243 ↾ cres 5521 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: fvsnun1 6921 |
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