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Theorem ovexg 5621
Description: Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
Assertion
Ref Expression
ovexg ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)

Proof of Theorem ovexg
StepHypRef Expression
1 df-ov 5597 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 simp2 942 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → 𝐹𝑊)
3 opexg 4022 . . . 4 ((𝐴𝑉𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
433adant2 960 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
5 fvexg 5272 . . 3 ((𝐹𝑊 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
62, 4, 5syl2anc 403 . 2 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
71, 6syl5eqel 2171 1 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 922  wcel 1436  Vcvv 2614  cop 3428  cfv 4972  (class class class)co 5594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-cnv 4412  df-dm 4414  df-rn 4415  df-iota 4937  df-fv 4980  df-ov 5597
This theorem is referenced by:  mapxpen  6497
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