Theorem funpartlem 30248

Description: Lemma for funpartfun 30249. 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 3065	. 2 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> A e. _V )
2		ssnid 3998	. . . . 5 |- x e. { x }
3		eleq2 2473	. . . . 5 |- ( ( F " { A } ) = { x } -> ( x e. ( F " { A } ) <-> x e. { x } ) )
4	2, 3	mpbiri 233	. . . 4 |- ( ( F " { A } ) = { x } -> x e. ( F " { A } ) )
5		n0i 3740	. . . . 5 |- ( x e. ( F " { A } ) -> -. ( F " { A } ) = (/) )
6		snprc 4032	. . . . . . . 8 |- ( -. A e. _V <-> { A } = (/) )
7	6	biimpi 194	. . . . . . 7 |- ( -. A e. _V -> { A } = (/) )
8	7	imaeq2d 5276	. . . . . 6 |- ( -. A e. _V -> ( F " { A } ) = ( F " (/) ) )
9		ima0 5291	. . . . . 6 |- ( F " (/) ) = (/)
10	8, 9	syl6eq 2457	. . . . 5 |- ( -. A e. _V -> ( F " { A } ) = (/) )
11	5, 10	nsyl2 127	. . . 4 |- ( x e. ( F " { A } ) -> A e. _V )
12	4, 11	syl 17	. . 3 |- ( ( F " { A } ) = { x } -> A e. _V )
13	12	exlimiv 1741	. 2 |- ( E. x ( F " { A } ) = { x } -> A e. _V )
14		eleq1 2472	. . 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 3979	. . . . . 6 |- ( y = A -> { y } = { A } )
16	15	imaeq2d 5276	. . . . 5 |- ( y = A -> ( F " { y } ) = ( F " { A } ) )
17	16	eqeq1d 2402	. . . 4 |- ( y = A -> ( ( F " { y } ) = { x } <-> ( F " { A } ) = { x } ) )
18	17	exbidv 1733	. . 3 |- ( y = A -> ( E. x ( F " { y } ) = { x } <-> E. x ( F " { A } ) = { x } ) )
19		vex 3059	. . . . 5 |- y e. _V
20	19	eldm 5140	. . . 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 4971	. . . . . . . . . 10 |- ( y ( _V X. Singletons ) z <-> ( y e. _V /\ z e. Singletons ) )
22	19, 21	mpbiran 917	. . . . . . . . 9 |- ( y ( _V X. Singletons ) z <-> z e. Singletons )
23		elsingles 30224	. . . . . . . . 9 |- ( z e. Singletons <-> E. x z = { x } )
24	22, 23	bitri 249	. . . . . . . 8 |- ( y ( _V X. Singletons ) z <-> E. x z = { x } )
25	24	anbi2i 692	. . . . . . 7 |- ( ( y (Image F o. Singleton ) z /\ y ( _V X. Singletons ) z ) <-> ( y (Image F o. Singleton ) z /\ E. x z = { x } ) )
26		brin 4441	. . . . . . 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 1797	. . . . . . 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 277	. . . . . 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 1686	. . . . 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 1871	. . . . 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 249	. . . 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 1690	. . . . . 6 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. z ( z = { x } /\ y (Image F o. Singleton ) z ) )
33		snex 4629	. . . . . . 7 |- { x } e. _V
34		breq2 4396	. . . . . . 7 |- ( z = { x } -> ( y (Image F o. Singleton ) z <-> y (Image F o. Singleton ) { x } ) )
35	33, 34	ceqsexv 3093	. . . . . 6 |- ( E. z ( z = { x } /\ y (Image F o. Singleton ) z ) <-> y (Image F o. Singleton ) { x } )
36	19, 33	brco 5113	. . . . . . 7 |- ( y (Image F o. Singleton ) { x } <-> E. z ( ySingleton z /\ zImage F { x } ) )
37		vex 3059	. . . . . . . . . 10 |- z e. _V
38	19, 37	brsingle 30223	. . . . . . . . 9 |- ( ySingleton z <-> z = { y } )
39	38	anbi1i 693	. . . . . . . 8 |- ( ( ySingleton z /\ zImage F { x } ) <-> ( z = { y } /\ zImage F { x } ) )
40	39	exbii 1686	. . . . . . 7 |- ( E. z ( ySingleton z /\ zImage F { x } ) <-> E. z ( z = { y } /\ zImage F { x } ) )
41		snex 4629	. . . . . . . . 9 |- { y } e. _V
42		breq1 4395	. . . . . . . . 9 |- ( z = { y } -> ( zImage F { x } <-> { y }Image F { x } ) )
43	41, 42	ceqsexv 3093	. . . . . . . 8 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> { y }Image F { x } )
44	41, 33	brimage 30232	. . . . . . . 8 |- ( { y }Image F { x } <-> { x } = ( F " { y } ) )
45		eqcom 2409	. . . . . . . 8 |- ( { x } = ( F " { y } ) <-> ( F " { y } ) = { x } )
46	43, 44, 45	3bitri 271	. . . . . . 7 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> ( F " { y } ) = { x } )
47	36, 40, 46	3bitri 271	. . . . . 6 |- ( y (Image F o. Singleton ) { x } <-> ( F " { y } ) = { x } )
48	32, 35, 47	3bitri 271	. . . . 5 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> ( F " { y } ) = { x } )
49	48	exbii 1686	. . . 4 |- ( E. x E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. x ( F " { y } ) = { x } )
50	20, 31, 49	3bitri 271	. . 3 |- ( y e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { y } ) = { x } )
51	14, 18, 50	vtoclbg 3115	. 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 351	1 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { A } ) = { x } )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 367   = wceq 1403   E.wex 1631   e. wcel 1840   _Vcvv 3056   i^i cin 3410   (/)c0 3735   {csn 3969   class class class wbr 4392   X. cxp 4938   dom cdm 4940   "cima 4943   o. ccom 4944  Singletoncsingle 30143   Singletonscsingles 30144  Imagecimage 30145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-symdif 3667  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-eprel 4731  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fo 5529  df-fv 5531  df-1st 6736  df-2nd 6737  df-txp 30159  df-singleton 30167  df-singles 30168  df-image 30169

This theorem is referenced by:  funpartfun  30249  funpartfv  30251