Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ovexg GIF version

Theorem ovexg 5539
 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 5515 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 simp2 905 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → 𝐹𝑊)
3 opexg 3964 . . . 4 ((𝐴𝑉𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
433adant2 923 . . 3 ((𝐴𝑉𝐹𝑊𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
5 fvexg 5194 . . 3 ((𝐹𝑊 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
62, 4, 5syl2anc 391 . 2 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ V)
71, 6syl5eqel 2124 1 ((𝐴𝑉𝐹𝑊𝐵𝑋) → (𝐴𝐹𝐵) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 885   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378  ‘cfv 4902  (class class class)co 5512 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator