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Mirrors > Home > MPE Home > Th. List > dmex | Structured version Visualization version GIF version |
Description: The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.) |
Ref | Expression |
---|---|
dmex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
dmex | ⊢ dom 𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | dmexg 6989 | . 2 ⊢ (𝐴 ∈ V → dom 𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ dom 𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: elxp4 7003 ofmres 7055 1stval 7061 fo1st 7079 frxp 7174 tfrlem8 7367 mapprc 7748 ixpprc 7815 bren 7850 brdomg 7851 ctex 7856 fundmen 7916 domssex 8006 mapen 8009 ssenen 8019 hartogslem1 8330 brwdomn0 8357 unxpwdom2 8376 ixpiunwdom 8379 oemapwe 8474 cantnffval2 8475 r0weon 8718 fseqenlem2 8731 acndom 8757 acndom2 8760 dfac9 8841 ackbij2lem2 8945 ackbij2lem3 8946 cfsmolem 8975 coftr 8978 dcomex 9152 axdc3lem4 9158 axdclem 9224 axdclem2 9225 fodomb 9229 brdom3 9231 brdom5 9232 brdom4 9233 hashfacen 13095 shftfval 13658 prdsval 15938 isoval 16248 issubc 16318 prfval 16662 symgbas 17623 psgnghm2 19746 dfac14 21231 indishmph 21411 ufldom 21576 tsmsval2 21743 dvmptadd 23529 dvmptmul 23530 dvmptco 23541 taylfval 23917 hmoval 27049 esum2d 29482 sitmval 29738 bnj893 30252 dfrecs2 31227 dfrdg4 31228 indexdom 32699 dibfval 35448 aomclem1 36642 dfac21 36654 trclexi 36946 rtrclexi 36947 dfrtrcl5 36955 dfrcl2 36985 dvsubf 38802 dvdivf 38812 fouriersw 39124 smflimlem1 39657 smflimlem6 39662 usgrsizedg 40442 usgredgaleordALT 40461 vtxdun 40696 vtxdlfgrval 40700 vtxd0nedgb 40703 vtxdushgrfvedglem 40704 vtxdushgrfvedg 40705 ewlksfval 40803 1wlksfval 40811 wlksfval 40812 1wlksv 40824 1wlkiswwlksupgr2 41074 vdn0conngrumgrv2 41363 vdgn1frgrv2 41466 |
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