Theorem funpartlem 29567

Description: Lemma for funpartfun 29568. 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 3104	. 2 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> A e. _V )
2		ssnid 4043	. . . . 5 |- x e. { x }
3		eleq2 2516	. . . . 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 3775	. . . . 5 |- ( x e. ( F " { A } ) -> -. ( F " { A } ) = (/) )
6		snprc 4078	. . . . . . . 8 |- ( -. A e. _V <-> { A } = (/) )
7	6	biimpi 194	. . . . . . 7 |- ( -. A e. _V -> { A } = (/) )
8	7	imaeq2d 5327	. . . . . 6 |- ( -. A e. _V -> ( F " { A } ) = ( F " (/) ) )
9		ima0 5342	. . . . . 6 |- ( F " (/) ) = (/)
10	8, 9	syl6eq 2500	. . . . 5 |- ( -. A e. _V -> ( F " { A } ) = (/) )
11	5, 10	nsyl2 127	. . . 4 |- ( x e. ( F " { A } ) -> A e. _V )
12	4, 11	syl 16	. . 3 |- ( ( F " { A } ) = { x } -> A e. _V )
13	12	exlimiv 1709	. 2 |- ( E. x ( F " { A } ) = { x } -> A e. _V )
14		eleq1 2515	. . 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 4024	. . . . . 6 |- ( y = A -> { y } = { A } )
16	15	imaeq2d 5327	. . . . 5 |- ( y = A -> ( F " { y } ) = ( F " { A } ) )
17	16	eqeq1d 2445	. . . 4 |- ( y = A -> ( ( F " { y } ) = { x } <-> ( F " { A } ) = { x } ) )
18	17	exbidv 1701	. . 3 |- ( y = A -> ( E. x ( F " { y } ) = { x } <-> E. x ( F " { A } ) = { x } ) )
19		vex 3098	. . . . 5 |- y e. _V
20	19	eldm 5190	. . . 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 5020	. . . . . . . . . 10 |- ( y ( _V X. Singletons ) z <-> ( y e. _V /\ z e. Singletons ) )
22	19, 21	mpbiran 918	. . . . . . . . 9 |- ( y ( _V X. Singletons ) z <-> z e. Singletons )
23		elsingles 29543	. . . . . . . . 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 694	. . . . . . 7 |- ( ( y (Image F o. Singleton ) z /\ y ( _V X. Singletons ) z ) <-> ( y (Image F o. Singleton ) z /\ E. x z = { x } ) )
26		brin 4486	. . . . . . 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 1761	. . . . . . 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 1654	. . . . 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 1835	. . . . 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 1658	. . . . . 6 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. z ( z = { x } /\ y (Image F o. Singleton ) z ) )
33		snex 4678	. . . . . . 7 |- { x } e. _V
34		breq2 4441	. . . . . . 7 |- ( z = { x } -> ( y (Image F o. Singleton ) z <-> y (Image F o. Singleton ) { x } ) )
35	33, 34	ceqsexv 3132	. . . . . 6 |- ( E. z ( z = { x } /\ y (Image F o. Singleton ) z ) <-> y (Image F o. Singleton ) { x } )
36	19, 33	brco 5163	. . . . . . 7 |- ( y (Image F o. Singleton ) { x } <-> E. z ( ySingleton z /\ zImage F { x } ) )
37		vex 3098	. . . . . . . . . 10 |- z e. _V
38	19, 37	brsingle 29542	. . . . . . . . 9 |- ( ySingleton z <-> z = { y } )
39	38	anbi1i 695	. . . . . . . 8 |- ( ( ySingleton z /\ zImage F { x } ) <-> ( z = { y } /\ zImage F { x } ) )
40	39	exbii 1654	. . . . . . 7 |- ( E. z ( ySingleton z /\ zImage F { x } ) <-> E. z ( z = { y } /\ zImage F { x } ) )
41		snex 4678	. . . . . . . . 9 |- { y } e. _V
42		breq1 4440	. . . . . . . . 9 |- ( z = { y } -> ( zImage F { x } <-> { y }Image F { x } ) )
43	41, 42	ceqsexv 3132	. . . . . . . 8 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> { y }Image F { x } )
44	41, 33	brimage 29551	. . . . . . . 8 |- ( { y }Image F { x } <-> { x } = ( F " { y } ) )
45		eqcom 2452	. . . . . . . 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 1654	. . . 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 3154	. 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 353	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 369   = wceq 1383   E.wex 1599   e. wcel 1804   _Vcvv 3095   i^i cin 3460   (/)c0 3770   {csn 4014   class class class wbr 4437   X. cxp 4987   dom cdm 4989   "cima 4992   o. ccom 4993  Singletoncsingle 29462   Singletonscsingles 29463  Imagecimage 29464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-eprel 4781  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-1st 6785  df-2nd 6786  df-symdif 29443  df-txp 29478  df-singleton 29486  df-singles 29487  df-image 29488

This theorem is referenced by:  funpartfun  29568  funpartfv  29570