![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6929. (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 6929 | . 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 2106 Vcvv 3474 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-dm 5686 df-iota 6495 df-fv 6551 |
This theorem is referenced by: mrieqv2d 17582 mreexmrid 17586 mreexexlem3d 17589 mreexexlem4d 17590 mreexexd 17591 mreexdomd 17592 acsdomd 18509 ismgmn0 18562 telgsumfz 19857 isirred 20232 tgclb 22472 alexsublem 23547 cnextcn 23570 ustssel 23709 fmucnd 23796 trcfilu 23798 cfiluweak 23799 ucnextcn 23808 imasdsf1olem 23878 imasf1oxmet 23880 comet 24021 restmetu 24078 wlkp1lem4 28930 wlkp1lem8 28934 1wlkdlem4 29390 eupth2lem3lem1 29478 eupth2lem3lem2 29479 gsumsubg 32193 opprqusplusg 32598 opprqus0g 32599 lbsdiflsp0 32706 fedgmullem1 32709 mzpcl34 41459 xlimbr 44533 xlimmnfvlem2 44539 xlimpnfvlem2 44543 ecqusaddcl 46759 |
Copyright terms: Public domain | W3C validator |