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Mirrors > Home > MPE Home > Th. List > dmresv | Structured version Visualization version GIF version |
Description: The domain of a universal restriction. (Contributed by NM, 14-May-2008.) |
Ref | Expression |
---|---|
dmresv | ⊢ dom (𝐴 ↾ V) = dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 5911 | . 2 ⊢ dom (𝐴 ↾ V) = (V ∩ dom 𝐴) | |
2 | incom 4140 | . 2 ⊢ (V ∩ dom 𝐴) = (dom 𝐴 ∩ V) | |
3 | inv1 4334 | . 2 ⊢ (dom 𝐴 ∩ V) = dom 𝐴 | |
4 | 1, 2, 3 | 3eqtri 2772 | 1 ⊢ dom (𝐴 ↾ V) = dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3431 ∩ cin 3891 dom cdm 5589 ↾ cres 5591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-xp 5595 df-dm 5599 df-res 5601 |
This theorem is referenced by: fidomdm 9072 dmttrcl 9455 dmct 10279 |
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