Theorem funopfv 5714

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 4110	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
2		funbrfv 5713	. 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))
3	1, 2	biimtrrid 153	1 ⊢ (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  ⟨cop 3692   class class class wbr 4109  Fun wfun 5346  ‘cfv 5352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by:  fvopab3ig  5751  fvsn  5879  ovidig  6171  ovigg  6174  f1o2ndf1  6424  fundmen  7047  frecuzrdg0  10775  frecuzrdgsuc  10776  frecuzrdg0t  10784  frecuzrdgsuctlem  10785  strslfvd  13254  strslfv2d  13255  imasaddvallemg  13528