Theorem fnex 5415

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 5414. (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 5028	. . 3 |- ( F Fn A -> Rel F )
2	1	adantr 270	. 2 |- ( ( F Fn A /\ A e. B ) -> Rel F )
3		df-fn 4935	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
4		eleq1a 2151	. . . . . 6 |- ( A e. B -> ( dom F = A -> dom F e. B ) )
5	4	impcom 123	. . . . 5 |- ( ( dom F = A /\ A e. B ) -> dom F e. B )
6		resfunexg 5414	. . . . 5 |- ( ( Fun F /\ dom F e. B ) -> ( F |` dom F ) e. _V )
7	5, 6	sylan2 280	. . . 4 |- ( ( Fun F /\ ( dom F = A /\ A e. B ) ) -> ( F |` dom F ) e. _V )
8	7	anassrs 392	. . 3 |- ( ( ( Fun F /\ dom F = A ) /\ A e. B ) -> ( F |` dom F ) e. _V )
9	3, 8	sylanb 278	. 2 |- ( ( F Fn A /\ A e. B ) -> ( F |` dom F ) e. _V )
10		resdm 4677	. . . 4 |- ( Rel F -> ( F |` dom F ) = F )
11	10	eleq1d 2148	. . 3 |- ( Rel F -> ( ( F |` dom F ) e. _V <-> F e. _V ) )
12	11	biimpa 290	. 2 |- ( ( Rel F /\ ( F |` dom F ) e. _V ) -> F e. _V )
13	2, 9, 12	syl2anc 403	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   _Vcvv 2602   dom cdm 4371   |` cres 4373   Rel wrel 4376   Fun wfun 4926   Fn wfn 4927

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940

This theorem is referenced by:  funex  5416  fex  5420  offval  5750  ofrfval  5751  tfrlemibex  5978  tfr1onlembex  5994  fndmeng  6357  frecfzennn  9508