Theorem fnex 5635

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 5634. (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 5216	. . 3 |- ( F Fn A -> Rel F )
2	1	adantr 274	. 2 |- ( ( F Fn A /\ A e. B ) -> Rel F )
3		df-fn 5121	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
4		eleq1a 2209	. . . . . 6 |- ( A e. B -> ( dom F = A -> dom F e. B ) )
5	4	impcom 124	. . . . 5 |- ( ( dom F = A /\ A e. B ) -> dom F e. B )
6		resfunexg 5634	. . . . 5 |- ( ( Fun F /\ dom F e. B ) -> ( F |` dom F ) e. _V )
7	5, 6	sylan2 284	. . . 4 |- ( ( Fun F /\ ( dom F = A /\ A e. B ) ) -> ( F |` dom F ) e. _V )
8	7	anassrs 397	. . 3 |- ( ( ( Fun F /\ dom F = A ) /\ A e. B ) -> ( F |` dom F ) e. _V )
9	3, 8	sylanb 282	. 2 |- ( ( F Fn A /\ A e. B ) -> ( F |` dom F ) e. _V )
10		resdm 4853	. . . 4 |- ( Rel F -> ( F |` dom F ) = F )
11	10	eleq1d 2206	. . 3 |- ( Rel F -> ( ( F |` dom F ) e. _V <-> F e. _V ) )
12	11	biimpa 294	. 2 |- ( ( Rel F /\ ( F |` dom F ) e. _V ) -> F e. _V )
13	2, 9, 12	syl2anc 408	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2681   dom cdm 4534   |` cres 4536   Rel wrel 4539   Fun wfun 5112   Fn wfn 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126

This theorem is referenced by:  funex  5636  fex  5640  offval  5982  ofrfval  5983  tfrlemibex  6219  tfr1onlembex  6235  fndmeng  6697  frecfzennn  10192