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Theorem nffn 5356
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 5274 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5293 . . 3  |-  F/ x Fun  F
42nfdm 4936 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2439 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1783 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1560 1  |-  F/ x  F  Fn  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   F/wnf 1534    = wceq 1632   F/_wnfc 2419   dom cdm 4705   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  nff  5403  nffo  5466  nfixp  6851  nfixp1  6852  feqmptdf  23243  stoweidlem31  27883  stoweidlem35  27887  stoweidlem59  27911  bnj1463  29401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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