Theorem funpartlem 30780

Description: Lemma for funpartfun 30781. 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 3040	. 2 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> A e. _V )
2		ssnid 3989	. . . . 5 |- x e. { x }
3		eleq2 2538	. . . . 5 |- ( ( F " { A } ) = { x } -> ( x e. ( F " { A } ) <-> x e. { x } ) )
4	2, 3	mpbiri 241	. . . 4 |- ( ( F " { A } ) = { x } -> x e. ( F " { A } ) )
5		n0i 3727	. . . . 5 |- ( x e. ( F " { A } ) -> -. ( F " { A } ) = (/) )
6		snprc 4027	. . . . . . . 8 |- ( -. A e. _V <-> { A } = (/) )
7	6	biimpi 199	. . . . . . 7 |- ( -. A e. _V -> { A } = (/) )
8	7	imaeq2d 5174	. . . . . 6 |- ( -. A e. _V -> ( F " { A } ) = ( F " (/) ) )
9		ima0 5189	. . . . . 6 |- ( F " (/) ) = (/)
10	8, 9	syl6eq 2521	. . . . 5 |- ( -. A e. _V -> ( F " { A } ) = (/) )
11	5, 10	nsyl2 132	. . . 4 |- ( x e. ( F " { A } ) -> A e. _V )
12	4, 11	syl 17	. . 3 |- ( ( F " { A } ) = { x } -> A e. _V )
13	12	exlimiv 1784	. 2 |- ( E. x ( F " { A } ) = { x } -> A e. _V )
14		eleq1 2537	. . 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 3969	. . . . . 6 |- ( y = A -> { y } = { A } )
16	15	imaeq2d 5174	. . . . 5 |- ( y = A -> ( F " { y } ) = ( F " { A } ) )
17	16	eqeq1d 2473	. . . 4 |- ( y = A -> ( ( F " { y } ) = { x } <-> ( F " { A } ) = { x } ) )
18	17	exbidv 1776	. . 3 |- ( y = A -> ( E. x ( F " { y } ) = { x } <-> E. x ( F " { A } ) = { x } ) )
19		vex 3034	. . . . 5 |- y e. _V
20	19	eldm 5037	. . . 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 4870	. . . . . . . . . 10 |- ( y ( _V X. Singletons ) z <-> ( y e. _V /\ z e. Singletons ) )
22	19, 21	mpbiran 932	. . . . . . . . 9 |- ( y ( _V X. Singletons ) z <-> z e. Singletons )
23		elsingles 30756	. . . . . . . . 9 |- ( z e. Singletons <-> E. x z = { x } )
24	22, 23	bitri 257	. . . . . . . 8 |- ( y ( _V X. Singletons ) z <-> E. x z = { x } )
25	24	anbi2i 708	. . . . . . 7 |- ( ( y (Image F o. Singleton ) z /\ y ( _V X. Singletons ) z ) <-> ( y (Image F o. Singleton ) z /\ E. x z = { x } ) )
26		brin 4445	. . . . . . 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 1842	. . . . . . 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 285	. . . . . 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 1726	. . . . 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 1944	. . . . 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 257	. . . 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 1730	. . . . . 6 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. z ( z = { x } /\ y (Image F o. Singleton ) z ) )
33		snex 4641	. . . . . . 7 |- { x } e. _V
34		breq2 4399	. . . . . . 7 |- ( z = { x } -> ( y (Image F o. Singleton ) z <-> y (Image F o. Singleton ) { x } ) )
35	33, 34	ceqsexv 3070	. . . . . 6 |- ( E. z ( z = { x } /\ y (Image F o. Singleton ) z ) <-> y (Image F o. Singleton ) { x } )
36	19, 33	brco 5010	. . . . . . 7 |- ( y (Image F o. Singleton ) { x } <-> E. z ( ySingleton z /\ zImage F { x } ) )
37		vex 3034	. . . . . . . . . 10 |- z e. _V
38	19, 37	brsingle 30755	. . . . . . . . 9 |- ( ySingleton z <-> z = { y } )
39	38	anbi1i 709	. . . . . . . 8 |- ( ( ySingleton z /\ zImage F { x } ) <-> ( z = { y } /\ zImage F { x } ) )
40	39	exbii 1726	. . . . . . 7 |- ( E. z ( ySingleton z /\ zImage F { x } ) <-> E. z ( z = { y } /\ zImage F { x } ) )
41		snex 4641	. . . . . . . . 9 |- { y } e. _V
42		breq1 4398	. . . . . . . . 9 |- ( z = { y } -> ( zImage F { x } <-> { y }Image F { x } ) )
43	41, 42	ceqsexv 3070	. . . . . . . 8 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> { y }Image F { x } )
44	41, 33	brimage 30764	. . . . . . . 8 |- ( { y }Image F { x } <-> { x } = ( F " { y } ) )
45		eqcom 2478	. . . . . . . 8 |- ( { x } = ( F " { y } ) <-> ( F " { y } ) = { x } )
46	43, 44, 45	3bitri 279	. . . . . . 7 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> ( F " { y } ) = { x } )
47	36, 40, 46	3bitri 279	. . . . . 6 |- ( y (Image F o. Singleton ) { x } <-> ( F " { y } ) = { x } )
48	32, 35, 47	3bitri 279	. . . . 5 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> ( F " { y } ) = { x } )
49	48	exbii 1726	. . . 4 |- ( E. x E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. x ( F " { y } ) = { x } )
50	20, 31, 49	3bitri 279	. . 3 |- ( y e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { y } ) = { x } )
51	14, 18, 50	vtoclbg 3094	. 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 360	1 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { A } ) = { x } )

Syntax hints:   -. wn 3   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   i^i cin 3389   (/)c0 3722   {csn 3959   class class class wbr 4395   X. cxp 4837   dom cdm 4839   "cima 4842   o. ccom 4843  Singletoncsingle 30675   Singletonscsingles 30676  Imagecimage 30677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691  df-singleton 30699  df-singles 30700  df-image 30701

This theorem is referenced by:  funpartfun  30781  funpartfv  30783