Theorem nffn 5472

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

Step	Hyp	Ref	Expression
1		df-fn 5375	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
2		nffn.1	. . . 4 |- F/_ x F
3	2	nffun 5395	. . 3 |- F/ x Fun F
4	2	nfdm 5021	. . . 4 |- F/_ x dom F
5		nffn.2	. . . 4 |- F/_ x A
6	4, 5	nfeq 2400	. . 3 |- F/ x dom F = A
7	3, 6	nfan 1618	. 2 |- F/ x ( Fun F /\ dom F = A )
8	1, 7	nfxfr 1527	1 |- F/ x F Fn A

Syntax hints:   /\ wa 104   = wceq 1402   F/wnf 1513   F/_wnfc 2379   dom cdm 4769   Fun wfun 5366   Fn wfn 5367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-fun 5374  df-fn 5375

This theorem is referenced by:  nff  5525  nffo  5609  nfixpxy  6989  nfixp1  6990