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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 5534 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐹‘𝐴) ∈ V) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐹‘𝐴) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 Vcvv 2737 ‘cfv 5216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-cnv 4634 df-dm 4636 df-rn 4637 df-iota 5178 df-fv 5224 |
This theorem is referenced by: rdgtfr 6374 rdgruledefgg 6375 mapsnf1o2 6695 ixpiinm 6723 mapsnen 6810 xpdom2 6830 mapxpen 6847 xpmapenlem 6848 phplem4 6854 ac6sfi 6897 fiintim 6927 acfun 7205 ccfunen 7262 ioof 9970 frec2uzrand 10404 frec2uzf1od 10405 frecfzennn 10425 hashinfom 10757 fsum3 11394 slotslfn 12487 ptex 12712 |
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