Theorem funopfv 6871

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 5090	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2		funbrfv 6870	. 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))
3	1, 2	biimtrrid 243	1 ⊢ (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4579   class class class wbr 5089  Fun wfun 6475  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by:  fvopab3ig  6925  fvsng  7114  fveqf1o  7236  ovidig  7488  ovigg  7491  funfv1st2nd  7978  funelss  7979  f1o2ndf1  8052  fundmen  8953  dif1en  9071  uzrdg0i  13866  uzrdgsuci  13867  strfvd  17111  strfv2d  17112  imasaddvallem  17433  imasvscafn  17441  noseqrdg0  28237  noseqrdgsuc  28238  adjeq  31915  bnj1379  34842  bnj97  34878  bnj553  34910  bnj966  34956  bnj1442  35061  satfv0fvfmla0  35457  satfv1fvfmla1  35467  nregmodellem  45119