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

Theorem funbrfvb 6940
Description: Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
Assertion
Ref Expression
funbrfvb ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))

Proof of Theorem funbrfvb
StepHypRef Expression
1 funfn 6572 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnbrfvb 6938 . 2 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
31, 2sylanb 580 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098   class class class wbr 5141  dom cdm 5669  Fun wfun 6531   Fn wfn 6532  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545
This theorem is referenced by:  funbrfv2b  6943  dfimafn  6948  funimass4  6950  eqfunresadj  7353  dcomex  10444  dvidlem  25799  taylthlem1  26263  dfimafnf  32369  funcnvmpt  32401  cantnf2  42648  frege124d  43085  frege129d  43087  ntrclsfv1  43379  ntrneifv1  43403  ntrneifv2  43404
  Copyright terms: Public domain W3C validator