Theorem fnbrovb 7441

Description: Value of a binary operation expressed as a binary relation. See also fnbrfvb 6914 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 7393	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	eqeq1i 2735	. 2 ⊢ ((𝐴𝐹𝐵) = 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) = 𝐶)
3		fnbrfvb2 6919	. 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 1540   ∈ wcel 2109  ⟨cop 4598   class class class wbr 5110   × cxp 5639   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522  df-ov 7393

This theorem is referenced by:  fnotovb  7442  sectpropdlem  49029  invpropdlem  49031  isopropdlem  49033