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Theorem fnbrfvb2 6954
Description: Version of fnbrfvb 6950 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 7469 for the form when 𝐹 is seen as a binary operation. (Contributed by BJ, 15-Feb-2022.)
Assertion
Ref Expression
fnbrfvb2 ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))

Proof of Theorem fnbrfvb2
StepHypRef Expression
1 opelxpi 5715 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑊))
2 fnbrfvb 6950 . 2 ((𝐹 Fn (𝑉 × 𝑊) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
31, 2sylan2 592 1 ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  cop 4635   class class class wbr 5148   × cxp 5676   Fn wfn 6543  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556
This theorem is referenced by:  fnbrovb  7469
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