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Theorem fvexg 5372
Description: Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
Assertion
Ref Expression
fvexg ((𝐹𝑉𝐴𝑊) → (𝐹𝐴) ∈ V)

Proof of Theorem fvexg
StepHypRef Expression
1 elex 2652 . . 3 (𝐴𝑊𝐴 ∈ V)
2 fvssunirng 5368 . . 3 (𝐴 ∈ V → (𝐹𝐴) ⊆ ran 𝐹)
31, 2syl 14 . 2 (𝐴𝑊 → (𝐹𝐴) ⊆ ran 𝐹)
4 rnexg 4740 . . 3 (𝐹𝑉 → ran 𝐹 ∈ V)
5 uniexg 4299 . . 3 (ran 𝐹 ∈ V → ran 𝐹 ∈ V)
64, 5syl 14 . 2 (𝐹𝑉 ran 𝐹 ∈ V)
7 ssexg 4007 . 2 (((𝐹𝐴) ⊆ ran 𝐹 ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
83, 6, 7syl2anr 286 1 ((𝐹𝑉𝐴𝑊) → (𝐹𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1448  Vcvv 2641  wss 3021   cuni 3683  ran crn 4478  cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-cnv 4485  df-dm 4487  df-rn 4488  df-iota 5024  df-fv 5067
This theorem is referenced by:  fvex  5373  ovexg  5737  rdgivallem  6208  frecabex  6225  mapsnconst  6518  addvalex  7531  uzennn  10050  absval  10613  climmpt  10908  strnfvnd  11761  ressid  11802  iscnp4  12168  cnpnei  12169
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