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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 7923 | . 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 2108 Vcvv 3480 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: fndmexd 7926 unxpwdom2 9628 wemapwe 9737 imadomg 10574 fpwwe2lem11 10681 fpwwe2lem12 10682 hashdmpropge2 14522 prdsplusg 17503 prdsmulr 17504 prdsvsca 17505 prdshom 17512 ssclem 17863 subsubc 17898 efgrcl 19733 dprdgrp 20025 dprdf 20026 dprdssv 20036 f1lindf 21842 decpmatval0 22770 pmatcollpw3lem 22789 ordtrest2lem 23211 ordtrest2 23212 mbfmulc2re 25683 mbfneg 25685 dvnf 25963 dvnbss 25964 dchrptlem3 27310 gsummpt2d 33052 gsumfs2d 33058 cycpmco2lem5 33150 cycpmconjslem2 33175 trclubgNEW 43631 omecl 46518 sssmf 46753 mbfresmf 46754 smfpimltxr 46762 smfpimgtxr 46795 smfres 46805 smfco 46817 |
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