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Theorem fvexg 5086
Description: Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
Assertion
Ref Expression
fvexg ((𝐹 𝑉 A 𝑊) → (𝐹A) V)

Proof of Theorem fvexg
StepHypRef Expression
1 elex 2541 . . 3 (A 𝑊A V)
2 fvssunirng 5082 . . 3 (A V → (𝐹A) ⊆ ran 𝐹)
31, 2syl 14 . 2 (A 𝑊 → (𝐹A) ⊆ ran 𝐹)
4 rnexg 4490 . . 3 (𝐹 𝑉 → ran 𝐹 V)
5 uniexg 4098 . . 3 (ran 𝐹 V → ran 𝐹 V)
64, 5syl 14 . 2 (𝐹 𝑉 ran 𝐹 V)
7 ssexg 3848 . 2 (((𝐹A) ⊆ ran 𝐹 ran 𝐹 V) → (𝐹A) V)
83, 6, 7syl2anr 274 1 ((𝐹 𝑉 A 𝑊) → (𝐹A) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  Vcvv 2533  wss 2895   cuni 3532  ran crn 4239  cfv 4796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-cnv 4246  df-dm 4248  df-rn 4249  df-iota 4761  df-fv 4804
This theorem is referenced by:  fvex  5087  rdgivallem  5853  frecabex  5865
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