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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 5558 | . 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
2 | inex1g 4935 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ dom 𝐴) ∈ V) | |
3 | 1, 2 | syl5eqel 2854 | 1 ⊢ (𝐵 ∈ 𝑉 → dom (𝐴 ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 Vcvv 3351 ∩ cin 3722 dom cdm 5249 ↾ cres 5251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-dm 5259 df-res 5261 |
This theorem is referenced by: resfunexg 6621 resfunexgALT 7274 |
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