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Theorem funopfv 6130
Description: The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
Assertion
Ref Expression
funopfv (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹𝐴) = 𝐵))

Proof of Theorem funopfv
StepHypRef Expression
1 df-br 4578 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2 funbrfv 6129 . 2 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
31, 2syl5bir 231 1 (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cop 4130   class class class wbr 4577  Fun wfun 5784  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798
This theorem is referenced by:  fvopab3ig  6173  fvsn  6329  fveqf1o  6435  ovidig  6654  ovigg  6657  f1o2ndf1  7149  fundmen  7893  uzrdg0i  12575  uzrdgsuci  12576  strfvd  15678  strfv2d  15679  imasaddvallem  15958  imasvscafn  15966  adjeq  27984  bnj1379  29961  bnj97  29996  bnj553  30028  bnj966  30074  bnj1442  30177  basvtxval  40244  edgfiedgval  40245
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