Theorem fnbrovb 7403

Description: Value of a binary operation expressed as a binary relation. See also fnbrfvb 6878 for functions on Cartesian products. (Contributed by BJ, 15-Feb-2022.)

Assertion

Ref	Expression
fnbrovb	⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ⟨𝐴, 𝐵⟩𝐹𝐶))

Proof of Theorem fnbrovb

Step	Hyp	Ref	Expression
1		df-ov 7355	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	eqeq1i 2738	. 2 ⊢ ((𝐴𝐹𝐵) = 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) = 𝐶)
3		fnbrfvb2 6883	. 2 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩𝐹𝐶))
4	2, 3	bitrid 283	1 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ⟨𝐴, 𝐵⟩𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4581   class class class wbr 5093   × cxp 5617   Fn wfn 6481  ‘cfv 6486  (class class class)co 7352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-ov 7355

This theorem is referenced by:  fnotovb  7404  sectpropdlem  49161  invpropdlem  49163  isopropdlem  49165