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

Theorem fnopfvb 6350
 Description: Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
Assertion
Ref Expression
fnopfvb ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Proof of Theorem fnopfvb
StepHypRef Expression
1 fnbrfvb 6349 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
2 df-br 4761 . 2 (𝐵𝐹𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹)
31, 2syl6bb 276 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1596   ∈ wcel 2103  ⟨cop 4291   class class class wbr 4760   Fn wfn 5996  ‘cfv 6001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fn 6004  df-fv 6009 This theorem is referenced by:  funopfvb  6352  fvopab3g  6391  f1ofveu  6760  fnotovbOLD  6811  ovid  6894  ov  6897  ovg  6916  wfrlem14  7548  tfrlem11  7604  rdglim2  7648  tz7.48-1  7658  mdetunilem9  20549
 Copyright terms: Public domain W3C validator