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Theorem funbrfvb 6709
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 6374 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnbrfvb 6707 . 2 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
31, 2sylanb 584 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115   class class class wbr 5053  dom cdm 5543  Fun wfun 6338   Fn wfn 6339  cfv 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fn 6347  df-fv 6352
This theorem is referenced by:  funbrfv2b  6712  dfimafn  6717  funimass4  6719  dcomex  9863  dvidlem  24516  taylthlem1  24966  dfimafnf  30387  funcnvmpt  30418  eqfunresadj  33031  frege124d  40318  frege129d  40320  ntrclsfv1  40617  ntrneifv1  40641  ntrneifv2  40642
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