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Theorem fnbrfvb2 6550
Description: Version of fnbrfvb 6546 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 7023 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 5441 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑊))
2 fnbrfvb 6546 . 2 ((𝐹 Fn (𝑉 × 𝑊) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
31, 2sylan2 584 1 ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴𝑉𝐵𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  cop 4442   class class class wbr 4926   × cxp 5402   Fn wfn 6181  cfv 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fn 6189  df-fv 6194
This theorem is referenced by:  fnbrovb  7023
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