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Theorem fnbrovb 7475
Description: Value of a binary operation expressed as a binary relation. See also fnbrfvb 6955 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.)
Assertion
Ref Expression
fnbrovb ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))

Proof of Theorem fnbrovb
StepHypRef Expression
1 df-ov 7429 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
21eqeq1i 2733 . 2 ((𝐴𝐹𝐵) = 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) = 𝐶)
3 fnbrfvb2 6959 . 2 ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
42, 3bitrid 282 1 ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cop 4638   class class class wbr 5152   × cxp 5680   Fn wfn 6548  cfv 6553  (class class class)co 7426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561  df-ov 7429
This theorem is referenced by:  fnotovb  7476
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