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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 7602 | . 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 2105 Vcvv 3492 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: unxpwdom2 9040 wemapwe 9148 imadomg 9944 fpwwe2lem12 10051 fpwwe2lem13 10052 hashdmpropge2 13829 prdsplusg 16719 prdsmulr 16720 prdsvsca 16721 prdshom 16728 ssclem 17077 subsubc 17111 efgrcl 18770 dprdgrp 19056 dprdf 19057 dprdssv 19067 f1lindf 20894 decpmatval0 21300 pmatcollpw3lem 21319 ordtrest2lem 21739 ordtrest2 21740 mbfmulc2re 24176 mbfneg 24178 dvnf 24451 dvnbss 24452 dchrptlem3 25769 gsummpt2d 30614 cycpmco2lem5 30699 cycpmconjslem2 30724 trclubgNEW 39856 omecl 42662 sssmf 42892 mbfresmf 42893 smfpimltxr 42901 smfpimgtxr 42933 smfres 42942 smfco 42954 isomgreqve 43867 |
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