Theorem fnexALT 7132

Description: Alternate proof of fnex 6481, derived using the Axiom of Replacement in the form of funimaexg 5975. This version uses ax-pow 4843 and ax-un 6949, whereas fnex 6481 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 5989	. . . 4 |- ( F Fn A -> Rel F )
2		relssdmrn 5656	. . . 4 |- ( Rel F -> F C_ ( dom F X. ran F ) )
3	1, 2	syl 17	. . 3 |- ( F Fn A -> F C_ ( dom F X. ran F ) )
4	3	adantr 481	. 2 |- ( ( F Fn A /\ A e. B ) -> F C_ ( dom F X. ran F ) )
5		fndm 5990	. . . . 5 |- ( F Fn A -> dom F = A )
6	5	eleq1d 2686	. . . 4 |- ( F Fn A -> ( dom F e. B <-> A e. B ) )
7	6	biimpar 502	. . 3 |- ( ( F Fn A /\ A e. B ) -> dom F e. B )
8		fnfun 5988	. . . . 5 |- ( F Fn A -> Fun F )
9		funimaexg 5975	. . . . 5 |- ( ( Fun F /\ A e. B ) -> ( F " A ) e. _V )
10	8, 9	sylan 488	. . . 4 |- ( ( F Fn A /\ A e. B ) -> ( F " A ) e. _V )
11		imadmrn 5476	. . . . . . 7 |- ( F " dom F ) = ran F
12	5	imaeq2d 5466	. . . . . . 7 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
13	11, 12	syl5eqr 2670	. . . . . 6 |- ( F Fn A -> ran F = ( F " A ) )
14	13	eleq1d 2686	. . . . 5 |- ( F Fn A -> ( ran F e. _V <-> ( F " A ) e. _V ) )
15	14	biimpar 502	. . . 4 |- ( ( F Fn A /\ ( F " A ) e. _V ) -> ran F e. _V )
16	10, 15	syldan 487	. . 3 |- ( ( F Fn A /\ A e. B ) -> ran F e. _V )
17		xpexg 6960	. . 3 |- ( ( dom F e. B /\ ran F e. _V ) -> ( dom F X. ran F ) e. _V )
18	7, 16, 17	syl2anc 693	. 2 |- ( ( F Fn A /\ A e. B ) -> ( dom F X. ran F ) e. _V )
19		ssexg 4804	. 2 |- ( ( F C_ ( dom F X. ran F ) /\ ( dom F X. ran F ) e. _V ) -> F e. _V )
20	4, 18, 19	syl2anc 693	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891

This theorem is referenced by: (None)