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Mirrors > Home > MPE Home > Th. List > dmresexg | Structured version Visualization version GIF version |
Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.) |
Ref | Expression |
---|---|
dmresexg | ⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 6021 | . 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
2 | inex1g 5323 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ dom 𝐴) ∈ V) | |
3 | 1, 2 | eqeltrid 2833 | 1 ⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3473 ∩ cin 3948 dom cdm 5682 ↾ 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-br 5153 df-opab 5215 df-xp 5688 df-dm 5692 df-res 5694 |
This theorem is referenced by: resfunexg 7233 resfunexgALT 7957 |
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