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Theorem funopfvb 6962
Description: Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
Assertion
Ref Expression
funopfvb ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Proof of Theorem funopfvb
StepHypRef Expression
1 funfn 6595 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnopfvb 6959 . 2 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
31, 2sylanb 581 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  cop 4631  dom cdm 5684  Fun wfun 6554   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  dmfco  7004  funfvop  7069  f1eqcocnv  7322  usgredgop  29188  fgreu  32683  gsumhashmul  33065
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