Theorem fnex 5759

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 5758. (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 5333	. . 3 |- ( F Fn A -> Rel F )
2	1	adantr 276	. 2 |- ( ( F Fn A /\ A e. B ) -> Rel F )
3		df-fn 5238	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
4		eleq1a 2261	. . . . . 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 5758	. . . . 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 4964	. . . 4 |- ( Rel F -> ( F |` dom F ) = F )
11	10	eleq1d 2258	. . 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 1364   e. wcel 2160   _Vcvv 2752   dom cdm 4644   |` cres 4646   Rel wrel 4649   Fun wfun 5229   Fn wfn 5230

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243

This theorem is referenced by:  funex  5760  fex  5766  offval  6115  ofrfval  6116  tfrlemibex  6355  tfr1onlembex  6371  fndmeng  6837  cc2lem  7296  frecfzennn  10459  xpscf  12826  mulgval  13079  mulgfng  13081  invrfvald  13489