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Theorem ovexg 5805
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 5777 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 simp2 982 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → 𝐹𝑊)
3 opexg 4150 . . . 4 ((𝐴𝑉𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
433adant2 1000 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
5 fvexg 5440 . . 3 ((𝐹𝑊 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
62, 4, 5syl2anc 408 . 2 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
71, 6eqeltrid 2226 1 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 962  wcel 1480  Vcvv 2686  cop 3530  cfv 5123  (class class class)co 5774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  mapxpen  6742  metrest  12689
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