Theorem fnexALT 5760

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 5003. This version of fnex 5404 uses ax-pow 3948 and ax-un 4188, whereas fnex 5404 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 5017	. . . 4 |- ( F Fn A -> Rel F )
2		relssdmrn 4861	. . . 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 270	. 2 |- ( ( F Fn A /\ A e. B ) -> F C_ ( dom F X. ran F ) )
5		fndm 5018	. . . . 5 |- ( F Fn A -> dom F = A )
6	5	eleq1d 2147	. . . 4 |- ( F Fn A -> ( dom F e. B <-> A e. B ) )
7	6	biimpar 291	. . 3 |- ( ( F Fn A /\ A e. B ) -> dom F e. B )
8		fnfun 5016	. . . . 5 |- ( F Fn A -> Fun F )
9		funimaexg 5003	. . . . 5 |- ( ( Fun F /\ A e. B ) -> ( F " A ) e. _V )
10	8, 9	sylan 277	. . . 4 |- ( ( F Fn A /\ A e. B ) -> ( F " A ) e. _V )
11		imadmrn 4698	. . . . . . 7 |- ( F " dom F ) = ran F
12	5	imaeq2d 4688	. . . . . . 7 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
13	11, 12	syl5eqr 2127	. . . . . 6 |- ( F Fn A -> ran F = ( F " A ) )
14	13	eleq1d 2147	. . . . 5 |- ( F Fn A -> ( ran F e. _V <-> ( F " A ) e. _V ) )
15	14	biimpar 291	. . . 4 |- ( ( F Fn A /\ ( F " A ) e. _V ) -> ran F e. _V )
16	10, 15	syldan 276	. . 3 |- ( ( F Fn A /\ A e. B ) -> ran F e. _V )
17		xpexg 4470	. . 3 |- ( ( dom F e. B /\ ran F e. _V ) -> ( dom F X. ran F ) e. _V )
18	7, 16, 17	syl2anc 403	. 2 |- ( ( F Fn A /\ A e. B ) -> ( dom F X. ran F ) e. _V )
19		ssexg 3917	. 2 |- ( ( F C_ ( dom F X. ran F ) /\ ( dom F X. ran F ) e. _V ) -> F e. _V )
20	4, 18, 19	syl2anc 403	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   _Vcvv 2601   C_ wss 2973   X. cxp 4361   dom cdm 4363   ran crn 4364   "cima 4366   Rel wrel 4368   Fun wfun 4916   Fn wfn 4917

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924  df-fn 4925

This theorem is referenced by: (None)