Theorem aceq3lem 8943

Description: Lemma for dfac3 8944. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypothesis

Ref	Expression
aceq3lem.1	|- F = ( w e. dom y |-> ( f ` { u | w y u } ) )

Assertion

Ref	Expression
aceq3lem	|- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) -> E. f ( f C_ y /\ f Fn dom y ) )

Distinct variable group:   x, y, z, w, u, f

Allowed substitution hints:   F( x, y, z, w, u, f)

Proof of Theorem aceq3lem

Dummy variables h g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- y e. _V
2	1	rnex 7100	. . . . 5 |- ran y e. _V
3	2	pwex 4848	. . . 4 |- ~P ran y e. _V
4		raleq 3138	. . . . 5 |- ( x = ~P ran y -> ( A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) <-> A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) ) )
5	4	exbidv 1850	. . . 4 |- ( x = ~P ran y -> ( E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) <-> E. f A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) ) )
6	3, 5	spcv 3299	. . 3 |- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) -> E. f A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) )
7		aceq3lem.1	. . . . . . 7 |- F = ( w e. dom y |-> ( f ` { u | w y u } ) )
8		df-mpt 4730	. . . . . . 7 |- ( w e. dom y |-> ( f ` { u | w y u } ) ) = { <. w , h >. | ( w e. dom y /\ h = ( f ` { u | w y u } ) ) }
9	7, 8	eqtri 2644	. . . . . 6 |- F = { <. w , h >. | ( w e. dom y /\ h = ( f ` { u | w y u } ) ) }
10		vex 3203	. . . . . . . . . . . . . . 15 |- w e. _V
11	10	eldm 5321	. . . . . . . . . . . . . 14 |- ( w e. dom y <-> E. u w y u )
12		abn0 3954	. . . . . . . . . . . . . 14 |- ( { u | w y u } =/= (/) <-> E. u w y u )
13	11, 12	bitr4i 267	. . . . . . . . . . . . 13 |- ( w e. dom y <-> { u | w y u } =/= (/) )
14		vex 3203	. . . . . . . . . . . . . . . . 17 |- u e. _V
15	10, 14	brelrn 5356	. . . . . . . . . . . . . . . 16 |- ( w y u -> u e. ran y )
16	15	abssi 3677	. . . . . . . . . . . . . . 15 |- { u | w y u } C_ ran y
17	2	elpw2 4828	. . . . . . . . . . . . . . 15 |- ( { u | w y u } e. ~P ran y <-> { u | w y u } C_ ran y )
18	16, 17	mpbir 221	. . . . . . . . . . . . . 14 |- { u | w y u } e. ~P ran y
19		neeq1 2856	. . . . . . . . . . . . . . . 16 |- ( z = { u | w y u } -> ( z =/= (/) <-> { u | w y u } =/= (/) ) )
20		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( z = { u | w y u } -> ( f ` z ) = ( f ` { u | w y u } ) )
21		id 22	. . . . . . . . . . . . . . . . 17 |- ( z = { u | w y u } -> z = { u | w y u } )
22	20, 21	eleq12d 2695	. . . . . . . . . . . . . . . 16 |- ( z = { u | w y u } -> ( ( f ` z ) e. z <-> ( f ` { u | w y u } ) e. { u | w y u } ) )
23	19, 22	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( z = { u | w y u } -> ( ( z =/= (/) -> ( f ` z ) e. z ) <-> ( { u | w y u } =/= (/) -> ( f ` { u | w y u } ) e. { u | w y u } ) ) )
24	23	rspcv 3305	. . . . . . . . . . . . . 14 |- ( { u | w y u } e. ~P ran y -> ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> ( { u | w y u } =/= (/) -> ( f ` { u | w y u } ) e. { u | w y u } ) ) )
25	18, 24	ax-mp 5	. . . . . . . . . . . . 13 |- ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> ( { u | w y u } =/= (/) -> ( f ` { u | w y u } ) e. { u | w y u } ) )
26	13, 25	syl5bi 232	. . . . . . . . . . . 12 |- ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> ( w e. dom y -> ( f ` { u | w y u } ) e. { u | w y u } ) )
27	26	imp 445	. . . . . . . . . . 11 |- ( ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) /\ w e. dom y ) -> ( f ` { u | w y u } ) e. { u | w y u } )
28		fvex 6201	. . . . . . . . . . . 12 |- ( f ` { u | w y u } ) e. _V
29		breq2 4657	. . . . . . . . . . . 12 |- ( z = ( f ` { u | w y u } ) -> ( w y z <-> w y ( f ` { u | w y u } ) ) )
30		breq2 4657	. . . . . . . . . . . . 13 |- ( u = z -> ( w y u <-> w y z ) )
31	30	cbvabv 2747	. . . . . . . . . . . 12 |- { u | w y u } = { z | w y z }
32	28, 29, 31	elab2 3354	. . . . . . . . . . 11 |- ( ( f ` { u | w y u } ) e. { u | w y u } <-> w y ( f ` { u | w y u } ) )
33	27, 32	sylib 208	. . . . . . . . . 10 |- ( ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) /\ w e. dom y ) -> w y ( f ` { u | w y u } ) )
34		breq2 4657	. . . . . . . . . 10 |- ( h = ( f ` { u | w y u } ) -> ( w y h <-> w y ( f ` { u | w y u } ) ) )
35	33, 34	syl5ibrcom 237	. . . . . . . . 9 |- ( ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) /\ w e. dom y ) -> ( h = ( f ` { u | w y u } ) -> w y h ) )
36	35	expimpd 629	. . . . . . . 8 |- ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> ( ( w e. dom y /\ h = ( f ` { u | w y u } ) ) -> w y h ) )
37	36	ssopab2dv 5004	. . . . . . 7 |- ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> { <. w , h >. | ( w e. dom y /\ h = ( f ` { u | w y u } ) ) } C_ { <. w , h >. | w y h } )
38		opabss 4714	. . . . . . 7 |- { <. w , h >. | w y h } C_ y
39	37, 38	syl6ss 3615	. . . . . 6 |- ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> { <. w , h >. | ( w e. dom y /\ h = ( f ` { u | w y u } ) ) } C_ y )
40	9, 39	syl5eqss 3649	. . . . 5 |- ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> F C_ y )
41	28, 7	fnmpti 6022	. . . . 5 |- F Fn dom y
42	1	ssex 4802	. . . . . . 7 |- ( F C_ y -> F e. _V )
43	42	adantr 481	. . . . . 6 |- ( ( F C_ y /\ F Fn dom y ) -> F e. _V )
44		sseq1 3626	. . . . . . . 8 |- ( g = F -> ( g C_ y <-> F C_ y ) )
45		fneq1 5979	. . . . . . . 8 |- ( g = F -> ( g Fn dom y <-> F Fn dom y ) )
46	44, 45	anbi12d 747	. . . . . . 7 |- ( g = F -> ( ( g C_ y /\ g Fn dom y ) <-> ( F C_ y /\ F Fn dom y ) ) )
47	46	spcegv 3294	. . . . . 6 |- ( F e. _V -> ( ( F C_ y /\ F Fn dom y ) -> E. g ( g C_ y /\ g Fn dom y ) ) )
48	43, 47	mpcom 38	. . . . 5 |- ( ( F C_ y /\ F Fn dom y ) -> E. g ( g C_ y /\ g Fn dom y ) )
49	40, 41, 48	sylancl 694	. . . 4 |- ( A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> E. g ( g C_ y /\ g Fn dom y ) )
50	49	exlimiv 1858	. . 3 |- ( E. f A. z e. ~P ran y ( z =/= (/) -> ( f ` z ) e. z ) -> E. g ( g C_ y /\ g Fn dom y ) )
51	6, 50	syl 17	. 2 |- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) -> E. g ( g C_ y /\ g Fn dom y ) )
52		sseq1 3626	. . . 4 |- ( g = f -> ( g C_ y <-> f C_ y ) )
53		fneq1 5979	. . . 4 |- ( g = f -> ( g Fn dom y <-> f Fn dom y ) )
54	52, 53	anbi12d 747	. . 3 |- ( g = f -> ( ( g C_ y /\ g Fn dom y ) <-> ( f C_ y /\ f Fn dom y ) ) )
55	54	cbvexv 2275	. 2 |- ( E. g ( g C_ y /\ g Fn dom y ) <-> E. f ( f C_ y /\ f Fn dom y ) )
56	51, 55	sylib 208	1 |- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) -> E. f ( f C_ y /\ f Fn dom y ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   {copab 4712   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   ` cfv 5888

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-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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  dfac3  8944