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Mirrors > Home > ILE Home > Th. List > fvex | GIF version |
Description: Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.) |
Ref | Expression |
---|---|
fvex.1 | ⊢ 𝐹 ∈ 𝑉 |
fvex.2 | ⊢ 𝐴 ∈ 𝑊 |
Ref | Expression |
---|---|
fvex | ⊢ (𝐹‘𝐴) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex.1 | . 2 ⊢ 𝐹 ∈ 𝑉 | |
2 | fvex.2 | . 2 ⊢ 𝐴 ∈ 𝑊 | |
3 | fvexg 5440 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐹‘𝐴) ∈ V) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ (𝐹‘𝐴) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 ‘cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-cnv 4547 df-dm 4549 df-rn 4550 df-iota 5088 df-fv 5131 |
This theorem is referenced by: rdgtfr 6271 rdgruledefgg 6272 mapsnf1o2 6590 ixpiinm 6618 mapsnen 6705 xpdom2 6725 mapxpen 6742 xpmapenlem 6743 phplem4 6749 ac6sfi 6792 fiintim 6817 acfun 7063 ccfunen 7079 ioof 9754 frec2uzrand 10178 frec2uzf1od 10179 frecfzennn 10199 hashinfom 10524 fsum3 11156 slotslfn 11985 |
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