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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, then the argument is a set. (An artifact of our function value definition.) (Contributed by Mario Carneiro, 6-Nov-2015.) |
Ref | Expression |
---|---|
elfvex | ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6938 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹) | |
2 | 1 | elexd 3485 | 1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3462 dom cdm 5682 ‘cfv 6554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-dm 5692 df-iota 6506 df-fv 6562 |
This theorem is referenced by: elfvexd 6940 fviss 6979 fiin 9465 elharval 9604 elfzp12 13634 ismre 17603 ismri 17644 isacs 17664 oppccofval 17730 mulgnngsum 19073 gexid 19579 efgrcl 19713 islss 20911 thlle 21694 thlleOLD 21695 islbs4 21830 istopon 22905 fgval 23865 fgcl 23873 ufilen 23925 ustssxp 24200 ustbasel 24202 ustincl 24203 ustdiag 24204 ustinvel 24205 ustexhalf 24206 ustfilxp 24208 ustbas2 24221 trust 24225 utopval 24228 elutop 24229 restutop 24233 ustuqtop5 24241 isucn 24274 psmetdmdm 24302 psmetf 24303 psmet0 24305 psmettri2 24306 psmetres2 24311 ismet2 24330 xmetpsmet 24345 metustfbas 24557 metust 24558 iscmet 25303 ulmscl 26408 1vgrex 28938 wlkcompim 29569 clwlkcompim 29717 wwlkbp 29775 2wlkdlem7 29866 clwwlkbp 29918 3wlkdlem7 30099 metidval 33705 pstmval 33710 pstmxmet 33712 issiga 33945 insiga 33970 mvrsval 35333 mrsubcv 35338 mrsubccat 35346 mppsval 35400 topdifinffinlem 37054 istotbnd 37470 isbnd 37481 ismrc 42358 isnacs 42361 mzpcl1 42386 mzpcl2 42387 mzpf 42393 mzpadd 42395 mzpmul 42396 mzpsubmpt 42400 mzpnegmpt 42401 mzpexpmpt 42402 mzpindd 42403 mzpsubst 42405 mzpcompact2 42409 mzpcong 42630 sprel 47056 clintop 47585 assintop 47586 clintopcllaw 47588 assintopcllaw 47589 assintopass 47591 |
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