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Mirrors > Home > MPE Home > Th. List > elfvexd | Structured version Visualization version GIF version |
Description: If a function value has a member, then its argument is a set. Deduction form of elfvex 6944. (An artifact of our function value definition.) (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
elfvexd.1 | ⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶)) |
Ref | Expression |
---|---|
elfvexd | ⊢ (𝜑 → 𝐶 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶)) | |
2 | elfvex 6944 | . 2 ⊢ (𝐴 ∈ (𝐵‘𝐶) → 𝐶 ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐶 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3477 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-dm 5698 df-iota 6515 df-fv 6570 |
This theorem is referenced by: mrieqv2d 17683 mreexmrid 17687 mreexexlem3d 17690 mreexexlem4d 17691 mreexexd 17692 mreexdomd 17693 acsdomd 18614 ismgmn0 18667 ecqusaddcl 19223 telgsumfz 20022 isirred 20435 tgclb 22992 alexsublem 24067 cnextcn 24090 ustssel 24229 fmucnd 24316 trcfilu 24318 cfiluweak 24319 ucnextcn 24328 imasdsf1olem 24398 imasf1oxmet 24400 comet 24541 restmetu 24598 wlkp1lem4 29708 wlkp1lem8 29712 1wlkdlem4 30168 eupth2lem3lem1 30256 eupth2lem3lem2 30257 gsumsubg 33031 opprqusplusg 33496 opprqus0g 33497 lsssra 33617 lbsdiflsp0 33653 fedgmullem1 33656 mzpcl34 42718 xlimbr 45782 xlimmnfvlem2 45788 xlimpnfvlem2 45792 |
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