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Mirrors > Home > MPE Home > Th. List > dmexd | Structured version Visualization version GIF version |
Description: The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
dmexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
dmexd | ⊢ (𝜑 → dom 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | dmexg 7909 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → dom 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3461 dom cdm 5678 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
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 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: fndmexd 7912 unxpwdom2 9613 wemapwe 9722 imadomg 10559 fpwwe2lem11 10666 fpwwe2lem12 10667 hashdmpropge2 14480 prdsplusg 17443 prdsmulr 17444 prdsvsca 17445 prdshom 17452 ssclem 17805 subsubc 17842 efgrcl 19682 dprdgrp 19974 dprdf 19975 dprdssv 19985 f1lindf 21773 decpmatval0 22710 pmatcollpw3lem 22729 ordtrest2lem 23151 ordtrest2 23152 mbfmulc2re 25621 mbfneg 25623 dvnf 25901 dvnbss 25902 dchrptlem3 27244 gsummpt2d 32853 cycpmco2lem5 32943 cycpmconjslem2 32968 trclubgNEW 43190 omecl 46029 sssmf 46264 mbfresmf 46265 smfpimltxr 46273 smfpimgtxr 46306 smfres 46316 smfco 46328 |
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