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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpdisjres | Structured version Visualization version GIF version |
Description: Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
xpdisjres | ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5694 | . 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
2 | xpdisj1 6170 | . 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ∅) | |
3 | 1, 2 | eqtrid 2780 | 1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3473 ∩ cin 3948 ∅c0 4326 × cxp 5680 ↾ cres 5684 |
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-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-opab 5215 df-xp 5688 df-rel 5689 df-res 5694 |
This theorem is referenced by: padct 32522 |
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