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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 6928. (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 6928 | . 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 2104 Vcvv 3472 ‘cfv 6542 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-dm 5685 df-iota 6494 df-fv 6550 |
This theorem is referenced by: mrieqv2d 17587 mreexmrid 17591 mreexexlem3d 17594 mreexexlem4d 17595 mreexexd 17596 mreexdomd 17597 acsdomd 18514 ismgmn0 18567 ecqusaddcl 19108 telgsumfz 19899 isirred 20310 tgclb 22693 alexsublem 23768 cnextcn 23791 ustssel 23930 fmucnd 24017 trcfilu 24019 cfiluweak 24020 ucnextcn 24029 imasdsf1olem 24099 imasf1oxmet 24101 comet 24242 restmetu 24299 wlkp1lem4 29200 wlkp1lem8 29204 1wlkdlem4 29660 eupth2lem3lem1 29748 eupth2lem3lem2 29749 gsumsubg 32468 opprqusplusg 32877 opprqus0g 32878 lsssra 32963 lbsdiflsp0 32999 fedgmullem1 33002 mzpcl34 41771 xlimbr 44841 xlimmnfvlem2 44847 xlimpnfvlem2 44851 |
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