Theorem funopfv 6821

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 → (𝐹‘𝐴) = 𝐵))

Proof of Theorem funopfv

Step	Hyp	Ref	Expression
1		df-br 5075	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2		funbrfv 6820	. 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))
3	1, 2	syl5bir 242	1 ⊢ (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  Fun wfun 6427  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by:  fvopab3ig  6871  fvsng  7052  fveqf1o  7175  ovidig  7415  ovigg  7418  funfv1st2nd  7887  funelss  7888  f1o2ndf1  7963  fundmen  8821  dif1en  8945  uzrdg0i  13679  uzrdgsuci  13680  strfvd  16902  strfv2d  16903  imasaddvallem  17240  imasvscafn  17248  adjeq  30297  bnj1379  32810  bnj97  32846  bnj553  32878  bnj966  32924  bnj1442  33029  satfv0fvfmla0  33375  satfv1fvfmla1  33385