MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funopfv Structured version   Visualization version   GIF version

Theorem funopfv 6958
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 5144 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2 funbrfv 6957 . 2 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹𝐴) = 𝐵))
31, 2biimtrrid 243 1 (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143  Fun wfun 6555  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  fvopab3ig  7012  fvsng  7200  fveqf1o  7322  ovidig  7575  ovigg  7578  funfv1st2nd  8071  funelss  8072  f1o2ndf1  8147  fundmen  9071  dif1en  9200  dif1enOLD  9202  uzrdg0i  14000  uzrdgsuci  14001  strfvd  17237  strfv2d  17238  imasaddvallem  17574  imasvscafn  17582  noseqrdg0  28313  noseqrdgsuc  28314  adjeq  31954  bnj1379  34844  bnj97  34880  bnj553  34912  bnj966  34958  bnj1442  35063  satfv0fvfmla0  35418  satfv1fvfmla1  35428
  Copyright terms: Public domain W3C validator