Theorem fnexALT 5740

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 4983. This version of fnex 5383 uses ax-pow 3927 and ax-un 4170, whereas fnex 5383 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 4997	. . . 4 |- ( F Fn A -> Rel F )
2		relssdmrn 4841	. . . 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 261	. 2 |- ( ( F Fn A /\ A e. B ) -> F C_ ( dom F X. ran F ) )
5		fndm 4998	. . . . 5 |- ( F Fn A -> dom F = A )
6	5	eleq1d 2106	. . . 4 |- ( F Fn A -> ( dom F e. B <-> A e. B ) )
7	6	biimpar 281	. . 3 |- ( ( F Fn A /\ A e. B ) -> dom F e. B )
8		fnfun 4996	. . . . 5 |- ( F Fn A -> Fun F )
9		funimaexg 4983	. . . . 5 |- ( ( Fun F /\ A e. B ) -> ( F " A ) e. _V )
10	8, 9	sylan 267	. . . 4 |- ( ( F Fn A /\ A e. B ) -> ( F " A ) e. _V )
11		imadmrn 4678	. . . . . . 7 |- ( F " dom F ) = ran F
12	5	imaeq2d 4668	. . . . . . 7 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
13	11, 12	syl5eqr 2086	. . . . . 6 |- ( F Fn A -> ran F = ( F " A ) )
14	13	eleq1d 2106	. . . . 5 |- ( F Fn A -> ( ran F e. _V <-> ( F " A ) e. _V ) )
15	14	biimpar 281	. . . 4 |- ( ( F Fn A /\ ( F " A ) e. _V ) -> ran F e. _V )
16	10, 15	syldan 266	. . 3 |- ( ( F Fn A /\ A e. B ) -> ran F e. _V )
17		xpexg 4452	. . 3 |- ( ( dom F e. B /\ ran F e. _V ) -> ( dom F X. ran F ) e. _V )
18	7, 16, 17	syl2anc 391	. 2 |- ( ( F Fn A /\ A e. B ) -> ( dom F X. ran F ) e. _V )
19		ssexg 3896	. 2 |- ( ( F C_ ( dom F X. ran F ) /\ ( dom F X. ran F ) e. _V ) -> F e. _V )
20	4, 18, 19	syl2anc 391	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   _Vcvv 2557   C_ wss 2917   X. cxp 4343   dom cdm 4345   ran crn 4346   "cima 4348   Rel wrel 4350   Fun wfun 4896   Fn wfn 4897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904  df-fn 4905

This theorem is referenced by: (None)