Theorem fnex 5739

Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5738. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fnex	|- ( ( F Fn A /\ A e. B ) -> F e. _V )

Proof of Theorem fnex

Step	Hyp	Ref	Expression
1		fnrel 5315	. . 3 |- ( F Fn A -> Rel F )
2	1	adantr 276	. 2 |- ( ( F Fn A /\ A e. B ) -> Rel F )
3		df-fn 5220	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
4		eleq1a 2249	. . . . . 6 |- ( A e. B -> ( dom F = A -> dom F e. B ) )
5	4	impcom 125	. . . . 5 |- ( ( dom F = A /\ A e. B ) -> dom F e. B )
6		resfunexg 5738	. . . . 5 |- ( ( Fun F /\ dom F e. B ) -> ( F |` dom F ) e. _V )
7	5, 6	sylan2 286	. . . 4 |- ( ( Fun F /\ ( dom F = A /\ A e. B ) ) -> ( F |` dom F ) e. _V )
8	7	anassrs 400	. . 3 |- ( ( ( Fun F /\ dom F = A ) /\ A e. B ) -> ( F |` dom F ) e. _V )
9	3, 8	sylanb 284	. 2 |- ( ( F Fn A /\ A e. B ) -> ( F |` dom F ) e. _V )
10		resdm 4947	. . . 4 |- ( Rel F -> ( F |` dom F ) = F )
11	10	eleq1d 2246	. . 3 |- ( Rel F -> ( ( F |` dom F ) e. _V <-> F e. _V ) )
12	11	biimpa 296	. 2 |- ( ( Rel F /\ ( F |` dom F ) e. _V ) -> F e. _V )
13	2, 9, 12	syl2anc 411	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2738   dom cdm 4627   |` cres 4629   Rel wrel 4632   Fun wfun 5211   Fn wfn 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225

This theorem is referenced by:  funex  5740  fex  5746  offval  6090  ofrfval  6091  tfrlemibex  6330  tfr1onlembex  6346  fndmeng  6810  cc2lem  7265  frecfzennn  10426  xpscf  12766  mulgval  12986  mulgfng  12987  invrfvald  13291