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Theorem nffn 5340
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 5258 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5277 . . 3  |-  F/ x Fun  F
42nfdm 4920 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2426 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1771 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1557 1  |-  F/ x  F  Fn  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   F/wnf 1531    = wceq 1623   F/_wnfc 2406   dom cdm 4689   Fun wfun 5249    Fn wfn 5250
This theorem is referenced by:  nff  5387  nffo  5450  nfixp  6835  nfixp1  6836  feqmptdf  23228  stoweidlem31  27780  stoweidlem35  27784  stoweidlem59  27808  bnj1463  29085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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