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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 2541 . . 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 3868 . 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 1374  Vcvv 2533  wss 2892   cuni 3552  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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899  ax-pr 3916  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-br 3737  df-opab 3791  df-cnv 4278  df-dm 4280  df-rn 4281  df-iota 4792  df-fv 4835
This theorem is referenced by:  fvex  5118  rdgivallem  5886  frecabex  5897
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