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Theorem nffn 5677
Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
Hypotheses
Ref Expression
nffn.1  |-  F/_ x F
nffn.2  |-  F/_ x A
Assertion
Ref Expression
nffn  |-  F/ x  F  Fn  A

Proof of Theorem nffn
StepHypRef Expression
1 df-fn 5591 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5610 . . 3  |-  F/ x Fun  F
42nfdm 5244 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2640 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1875 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1625 1  |-  F/ x  F  Fn  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   F/wnf 1599   F/_wnfc 2615   dom cdm 4999   Fun wfun 5582    Fn wfn 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5590  df-fn 5591
This theorem is referenced by:  nff  5727  nffo  5794  nfixp  7488  nfixp1  7489  feqmptdf  27201  stoweidlem31  31359  stoweidlem35  31363  stoweidlem59  31387  bnj1463  33208
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