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Mirrors > Home > MPE Home > Th. List > elfvex | Structured version Visualization version GIF version |
Description: If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.) |
Ref | Expression |
---|---|
elfvex | ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6130 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹) | |
2 | elex 3185 | . 2 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 dom cdm 5038 ‘cfv 5804 |
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-nul 4717 ax-pow 4769 |
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-ne 2782 df-ral 2901 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-dm 5048 df-iota 5768 df-fv 5812 |
This theorem is referenced by: elfvexd 6132 fviss 6166 fiin 8211 elharval 8351 elfzp12 12288 ismre 16073 ismri 16114 isacs 16135 oppccofval 16199 gexid 17819 efgrcl 17951 islss 18756 thlle 19860 islbs4 19990 istopon 20540 fgval 21484 fgcl 21492 ufilen 21544 ustssxp 21818 ustbasel 21820 ustincl 21821 ustdiag 21822 ustinvel 21823 ustexhalf 21824 ustfilxp 21826 ustbas2 21839 trust 21843 utopval 21846 elutop 21847 restutop 21851 ustuqtop5 21859 isucn 21892 psmetdmdm 21920 psmetf 21921 psmet0 21923 psmettri2 21924 psmetres2 21929 ismet2 21948 xmetpsmet 21963 metustfbas 22172 metust 22173 iscmet 22890 ulmscl 23937 1vgrex 25679 metidval 29261 pstmval 29266 pstmxmet 29268 issiga 29501 insiga 29527 mvrsval 30656 mrsubcv 30661 mrsubccat 30669 mppsval 30723 topdifinffinlem 32371 istotbnd 32738 isbnd 32749 ismrc 36282 isnacs 36285 mzpcl1 36310 mzpcl2 36311 mzpf 36317 mzpadd 36319 mzpmul 36320 mzpsubmpt 36324 mzpnegmpt 36325 mzpexpmpt 36326 mzpindd 36327 mzpsubst 36329 mzpcompact2 36333 mzpcong 36557 uvtxaisvtx 40615 vtxnbuvtx 40617 uvtxanbgrvtx 40619 uvtxnbgr 40627 1wlkcompim 40836 usgr2wlkspth 40965 clwlk1wlk 40982 clWlkcompim 40987 wwlkbp 41043 21wlkdlem7 41139 clwwlkbp 41191 31wlkdlem7 41333 clintop 41634 assintop 41635 clintopcllaw 41637 assintopcllaw 41638 assintopass 41640 |
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