Theorem funopfv 6273

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 4686	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2		funbrfv 6272	. 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))
3	1, 2	syl5bir 233	1 ⊢ (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ⟨cop 4216   class class class wbr 4685  Fun wfun 5920  ‘cfv 5926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934

This theorem is referenced by:  fvopab3ig  6317  fvsn  6487  fveqf1o  6597  ovidig  6820  ovigg  6823  f1o2ndf1  7330  fundmen  8071  uzrdg0i  12798  uzrdgsuci  12799  strfvd  15951  strfv2d  15952  imasaddvallem  16236  imasvscafn  16244  basvtxvalOLD  25948  edgfiedgvalOLD  25949  adjeq  28922  bnj1379  31027  bnj97  31062  bnj553  31094  bnj966  31140  bnj1442  31243