Theorem fnexALT 5942

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 funimaexg 5143. This version of fnex 5574 uses ax-pow 4038 and ax-un 4293, whereas fnex 5574 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem fnexALT

Step	Hyp	Ref	Expression
1		fnrel 5157	. . . 4 |- ( F Fn A -> Rel F )
2		relssdmrn 4995	. . . 4 |- ( Rel F -> F C_ ( dom F X. ran F ) )
3	1, 2	syl 14	. . 3 |- ( F Fn A -> F C_ ( dom F X. ran F ) )
4	3	adantr 272	. 2 |- ( ( F Fn A /\ A e. B ) -> F C_ ( dom F X. ran F ) )
5		fndm 5158	. . . . 5 |- ( F Fn A -> dom F = A )
6	5	eleq1d 2168	. . . 4 |- ( F Fn A -> ( dom F e. B <-> A e. B ) )
7	6	biimpar 293	. . 3 |- ( ( F Fn A /\ A e. B ) -> dom F e. B )
8		fnfun 5156	. . . . 5 |- ( F Fn A -> Fun F )
9		funimaexg 5143	. . . . 5 |- ( ( Fun F /\ A e. B ) -> ( F " A ) e. _V )
10	8, 9	sylan 279	. . . 4 |- ( ( F Fn A /\ A e. B ) -> ( F " A ) e. _V )
11		imadmrn 4827	. . . . . . 7 |- ( F " dom F ) = ran F
12	5	imaeq2d 4817	. . . . . . 7 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
13	11, 12	syl5eqr 2146	. . . . . 6 |- ( F Fn A -> ran F = ( F " A ) )
14	13	eleq1d 2168	. . . . 5 |- ( F Fn A -> ( ran F e. _V <-> ( F " A ) e. _V ) )
15	14	biimpar 293	. . . 4 |- ( ( F Fn A /\ ( F " A ) e. _V ) -> ran F e. _V )
16	10, 15	syldan 278	. . 3 |- ( ( F Fn A /\ A e. B ) -> ran F e. _V )
17		xpexg 4591	. . 3 |- ( ( dom F e. B /\ ran F e. _V ) -> ( dom F X. ran F ) e. _V )
18	7, 16, 17	syl2anc 406	. 2 |- ( ( F Fn A /\ A e. B ) -> ( dom F X. ran F ) e. _V )
19		ssexg 4007	. 2 |- ( ( F C_ ( dom F X. ran F ) /\ ( dom F X. ran F ) e. _V ) -> F e. _V )
20	4, 18, 19	syl2anc 406	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1448   _Vcvv 2641   C_ wss 3021   X. cxp 4475   dom cdm 4477   ran crn 4478   "cima 4480   Rel wrel 4482   Fun wfun 5053   Fn wfn 5054

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-fun 5061  df-fn 5062

This theorem is referenced by: (None)