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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 6943 | . 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹) | |
| 2 | 1 | elexd 3504 | 1 ⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 dom cdm 5685 ‘cfv 6561 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-dm 5695 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: elfvexd 6945 fviss 6986 fiin 9462 elharval 9601 elfzp12 13643 ismre 17633 ismri 17674 isacs 17694 oppccofval 17759 mulgnngsum 19097 gexid 19599 efgrcl 19733 islss 20932 thlle 21716 thlleOLD 21717 islbs4 21852 istopon 22918 fgval 23878 fgcl 23886 ufilen 23938 ustssxp 24213 ustbasel 24215 ustincl 24216 ustdiag 24217 ustinvel 24218 ustexhalf 24219 ustfilxp 24221 ustbas2 24234 trust 24238 utopval 24241 elutop 24242 restutop 24246 ustuqtop5 24254 isucn 24287 psmetdmdm 24315 psmetf 24316 psmet0 24318 psmettri2 24319 psmetres2 24324 ismet2 24343 xmetpsmet 24358 metustfbas 24570 metust 24571 iscmet 25318 ulmscl 26422 1vgrex 29019 wlkcompim 29650 clwlkcompim 29800 wwlkbp 29861 2wlkdlem7 29952 clwwlkbp 30004 3wlkdlem7 30185 metidval 33889 pstmval 33894 pstmxmet 33896 issiga 34113 insiga 34138 mvrsval 35510 mrsubcv 35515 mrsubccat 35523 mppsval 35577 topdifinffinlem 37348 istotbnd 37776 isbnd 37787 ismrc 42712 isnacs 42715 mzpcl1 42740 mzpcl2 42741 mzpf 42747 mzpadd 42749 mzpmul 42750 mzpsubmpt 42754 mzpnegmpt 42755 mzpexpmpt 42756 mzpindd 42757 mzpsubst 42759 mzpcompact2 42763 mzpcong 42984 sprel 47471 grtriprop 47908 clintop 48124 assintop 48125 clintopcllaw 48127 assintopcllaw 48128 assintopass 48130 |
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