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Theorem ovexg 5884
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 5853 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 simp2 993 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → 𝐹𝑊)
3 opexg 4211 . . . 4 ((𝐴𝑉𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
433adant2 1011 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
5 fvexg 5513 . . 3 ((𝐹𝑊 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
62, 4, 5syl2anc 409 . 2 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
71, 6eqeltrid 2257 1 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 973  wcel 2141  Vcvv 2730  cop 3584  cfv 5196  (class class class)co 5850
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-cnv 4617  df-dm 4619  df-rn 4620  df-iota 5158  df-fv 5204  df-ov 5853
This theorem is referenced by:  mapxpen  6822  plusfvalg  12604  plusffng  12606  metrest  13259
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