Theorem fnex 5931

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 5930. (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 5477	. . 3 |- ( F Fn A -> Rel F )
2	1	adantr 276	. 2 |- ( ( F Fn A /\ A e. B ) -> Rel F )
3		df-fn 5378	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
4		eleq1a 2310	. . . . . 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 5930	. . . . 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 404	. . 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 5100	. . . 4 |- ( Rel F -> ( F |` dom F ) = F )
11	10	eleq1d 2307	. . 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 415	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   dom cdm 4772   |` cres 4774   Rel wrel 4777   Fun wfun 5369   Fn wfn 5370

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-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383

This theorem is referenced by:  fndmexb  5932  funex  5934  fex  5940  offval  6303  ofrfval  6304  uchoice  6364  suppvalfn  6474  suppfnss  6490  tfrlemibex  6593  tfr1onlembex  6609  fndmeng  7091  cc2lem  7625  frecfzennn  10844  xpscf  13648  mulgval  13905  mulgfng  13907  prdsbas2  14159  prdsplusgval  14163  prdsplusgfval  14164  prdsmulrval  14165  prdsmulrfval  14166  invrfvald  14405