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Theorem funbrfvb 6902
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 6536 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fnbrfvb 6900 . 2 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
31, 2sylanb 582 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5110  dom cdm 5638  Fun wfun 6495   Fn wfn 6496  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by:  funbrfv2b  6905  dfimafn  6910  funimass4  6912  eqfunresadj  7310  dcomex  10390  dvidlem  25295  taylthlem1  25748  dfimafnf  31592  funcnvmpt  31625  cantnf2  41689  frege124d  42107  frege129d  42109  ntrclsfv1  42401  ntrneifv1  42425  ntrneifv2  42426
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