Theorem funpartlem 32049

Description: Lemma for funpartfun 32050. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)

Assertion

Ref	Expression
funpartlem	|- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { A } ) = { x } )

Distinct variable groups:   x, A   x, F

Proof of Theorem funpartlem

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> A e. _V )
2		vsnid 4209	. . . . 5 |- x e. { x }
3		eleq2 2690	. . . . 5 |- ( ( F " { A } ) = { x } -> ( x e. ( F " { A } ) <-> x e. { x } ) )
4	2, 3	mpbiri 248	. . . 4 |- ( ( F " { A } ) = { x } -> x e. ( F " { A } ) )
5		n0i 3920	. . . . 5 |- ( x e. ( F " { A } ) -> -. ( F " { A } ) = (/) )
6		snprc 4253	. . . . . . . 8 |- ( -. A e. _V <-> { A } = (/) )
7	6	biimpi 206	. . . . . . 7 |- ( -. A e. _V -> { A } = (/) )
8	7	imaeq2d 5466	. . . . . 6 |- ( -. A e. _V -> ( F " { A } ) = ( F " (/) ) )
9		ima0 5481	. . . . . 6 |- ( F " (/) ) = (/)
10	8, 9	syl6eq 2672	. . . . 5 |- ( -. A e. _V -> ( F " { A } ) = (/) )
11	5, 10	nsyl2 142	. . . 4 |- ( x e. ( F " { A } ) -> A e. _V )
12	4, 11	syl 17	. . 3 |- ( ( F " { A } ) = { x } -> A e. _V )
13	12	exlimiv 1858	. 2 |- ( E. x ( F " { A } ) = { x } -> A e. _V )
14		eleq1 2689	. . 3 |- ( y = A -> ( y e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) )
15		sneq 4187	. . . . . 6 |- ( y = A -> { y } = { A } )
16	15	imaeq2d 5466	. . . . 5 |- ( y = A -> ( F " { y } ) = ( F " { A } ) )
17	16	eqeq1d 2624	. . . 4 |- ( y = A -> ( ( F " { y } ) = { x } <-> ( F " { A } ) = { x } ) )
18	17	exbidv 1850	. . 3 |- ( y = A -> ( E. x ( F " { y } ) = { x } <-> E. x ( F " { A } ) = { x } ) )
19		vex 3203	. . . . 5 |- y e. _V
20	19	eldm 5321	. . . 4 |- ( y e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. z y ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) z )
21		brxp 5147	. . . . . . . . . 10 |- ( y ( _V X. Singletons ) z <-> ( y e. _V /\ z e. Singletons ) )
22	19, 21	mpbiran 953	. . . . . . . . 9 |- ( y ( _V X. Singletons ) z <-> z e. Singletons )
23		elsingles 32025	. . . . . . . . 9 |- ( z e. Singletons <-> E. x z = { x } )
24	22, 23	bitri 264	. . . . . . . 8 |- ( y ( _V X. Singletons ) z <-> E. x z = { x } )
25	24	anbi2i 730	. . . . . . 7 |- ( ( y (Image F o. Singleton ) z /\ y ( _V X. Singletons ) z ) <-> ( y (Image F o. Singleton ) z /\ E. x z = { x } ) )
26		brin 4704	. . . . . . 7 |- ( y ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) z <-> ( y (Image F o. Singleton ) z /\ y ( _V X. Singletons ) z ) )
27		19.42v 1918	. . . . . . 7 |- ( E. x ( y (Image F o. Singleton ) z /\ z = { x } ) <-> ( y (Image F o. Singleton ) z /\ E. x z = { x } ) )
28	25, 26, 27	3bitr4i 292	. . . . . 6 |- ( y ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) z <-> E. x ( y (Image F o. Singleton ) z /\ z = { x } ) )
29	28	exbii 1774	. . . . 5 |- ( E. z y ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) z <-> E. z E. x ( y (Image F o. Singleton ) z /\ z = { x } ) )
30		excom 2042	. . . . 5 |- ( E. z E. x ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. x E. z ( y (Image F o. Singleton ) z /\ z = { x } ) )
31	29, 30	bitri 264	. . . 4 |- ( E. z y ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) z <-> E. x E. z ( y (Image F o. Singleton ) z /\ z = { x } ) )
32		exancom 1787	. . . . . 6 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. z ( z = { x } /\ y (Image F o. Singleton ) z ) )
33		snex 4908	. . . . . . 7 |- { x } e. _V
34		breq2 4657	. . . . . . 7 |- ( z = { x } -> ( y (Image F o. Singleton ) z <-> y (Image F o. Singleton ) { x } ) )
35	33, 34	ceqsexv 3242	. . . . . 6 |- ( E. z ( z = { x } /\ y (Image F o. Singleton ) z ) <-> y (Image F o. Singleton ) { x } )
36	19, 33	brco 5292	. . . . . . 7 |- ( y (Image F o. Singleton ) { x } <-> E. z ( ySingleton z /\ zImage F { x } ) )
37		vex 3203	. . . . . . . . . 10 |- z e. _V
38	19, 37	brsingle 32024	. . . . . . . . 9 |- ( ySingleton z <-> z = { y } )
39	38	anbi1i 731	. . . . . . . 8 |- ( ( ySingleton z /\ zImage F { x } ) <-> ( z = { y } /\ zImage F { x } ) )
40	39	exbii 1774	. . . . . . 7 |- ( E. z ( ySingleton z /\ zImage F { x } ) <-> E. z ( z = { y } /\ zImage F { x } ) )
41		snex 4908	. . . . . . . . 9 |- { y } e. _V
42		breq1 4656	. . . . . . . . 9 |- ( z = { y } -> ( zImage F { x } <-> { y }Image F { x } ) )
43	41, 42	ceqsexv 3242	. . . . . . . 8 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> { y }Image F { x } )
44	41, 33	brimage 32033	. . . . . . . 8 |- ( { y }Image F { x } <-> { x } = ( F " { y } ) )
45		eqcom 2629	. . . . . . . 8 |- ( { x } = ( F " { y } ) <-> ( F " { y } ) = { x } )
46	43, 44, 45	3bitri 286	. . . . . . 7 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> ( F " { y } ) = { x } )
47	36, 40, 46	3bitri 286	. . . . . 6 |- ( y (Image F o. Singleton ) { x } <-> ( F " { y } ) = { x } )
48	32, 35, 47	3bitri 286	. . . . 5 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> ( F " { y } ) = { x } )
49	48	exbii 1774	. . . 4 |- ( E. x E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. x ( F " { y } ) = { x } )
50	20, 31, 49	3bitri 286	. . 3 |- ( y e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { y } ) = { x } )
51	14, 18, 50	vtoclbg 3267	. 2 |- ( A e. _V -> ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { A } ) = { x } ) )
52	1, 13, 51	pm5.21nii 368	1 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { A } ) = { x } )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   dom cdm 5114   "cima 5117   o. ccom 5118  Singletoncsingle 31945   Singletonscsingles 31946  Imagecimage 31947

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-symdif 3844  df-nul 3916  df-if 4087  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-eprel 5029  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969  df-singles 31970  df-image 31971

This theorem is referenced by:  funpartfun  32050  funpartfv  32052