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Theorem fnbrovb 7185
 Description: Value of a binary operation expressed as a binary relation. See also fnbrfvb 6694 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 7139 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
21eqeq1i 2803 . 2 ((𝐴𝐹𝐵) = 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) = 𝐶)
3 fnbrfvb2 6698 . 2 ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
42, 3syl5bb 286 1 ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5031   × cxp 5518   Fn wfn 6320  ‘cfv 6325  (class class class)co 7136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fn 6328  df-fv 6333  df-ov 7139 This theorem is referenced by:  fnotovb  7186
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