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Theorem fvexg 5117
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 2542 . . 3 (A 𝑊A V)
2 fvssunirng 5113 . . 3 (A V → (𝐹A) ⊆ ran 𝐹)
31, 2syl 14 . 2 (A 𝑊 → (𝐹A) ⊆ ran 𝐹)
4 rnexg 4522 . . 3 (𝐹 𝑉 → ran 𝐹 V)
5 uniexg 4123 . . 3 (ran 𝐹 V → ran 𝐹 V)
64, 5syl 14 . 2 (𝐹 𝑉 ran 𝐹 V)
7 ssexg 3869 . 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 2534  wss 2893   cuni 3553  ran crn 4271  cfv 4827
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  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 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-br 3738  df-opab 3792  df-cnv 4278  df-dm 4280  df-rn 4281  df-iota 4792  df-fv 4835
This theorem is referenced by:  fvex  5118  rdgivallem  5883  frecabex  5894
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