Theorem funpartlem 30714

Description: Lemma for funpartfun 30715. 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 3089	. 2 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> A e. _V )
2		ssnid 4027	. . . . 5 |- x e. { x }
3		eleq2 2496	. . . . 5 |- ( ( F " { A } ) = { x } -> ( x e. ( F " { A } ) <-> x e. { x } ) )
4	2, 3	mpbiri 236	. . . 4 |- ( ( F " { A } ) = { x } -> x e. ( F " { A } ) )
5		n0i 3766	. . . . 5 |- ( x e. ( F " { A } ) -> -. ( F " { A } ) = (/) )
6		snprc 4063	. . . . . . . 8 |- ( -. A e. _V <-> { A } = (/) )
7	6	biimpi 197	. . . . . . 7 |- ( -. A e. _V -> { A } = (/) )
8	7	imaeq2d 5187	. . . . . 6 |- ( -. A e. _V -> ( F " { A } ) = ( F " (/) ) )
9		ima0 5202	. . . . . 6 |- ( F " (/) ) = (/)
10	8, 9	syl6eq 2479	. . . . 5 |- ( -. A e. _V -> ( F " { A } ) = (/) )
11	5, 10	nsyl2 130	. . . 4 |- ( x e. ( F " { A } ) -> A e. _V )
12	4, 11	syl 17	. . 3 |- ( ( F " { A } ) = { x } -> A e. _V )
13	12	exlimiv 1770	. 2 |- ( E. x ( F " { A } ) = { x } -> A e. _V )
14		eleq1 2495	. . 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 4008	. . . . . 6 |- ( y = A -> { y } = { A } )
16	15	imaeq2d 5187	. . . . 5 |- ( y = A -> ( F " { y } ) = ( F " { A } ) )
17	16	eqeq1d 2424	. . . 4 |- ( y = A -> ( ( F " { y } ) = { x } <-> ( F " { A } ) = { x } ) )
18	17	exbidv 1762	. . 3 |- ( y = A -> ( E. x ( F " { y } ) = { x } <-> E. x ( F " { A } ) = { x } ) )
19		vex 3083	. . . . 5 |- y e. _V
20	19	eldm 5051	. . . 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 4884	. . . . . . . . . 10 |- ( y ( _V X. Singletons ) z <-> ( y e. _V /\ z e. Singletons ) )
22	19, 21	mpbiran 926	. . . . . . . . 9 |- ( y ( _V X. Singletons ) z <-> z e. Singletons )
23		elsingles 30690	. . . . . . . . 9 |- ( z e. Singletons <-> E. x z = { x } )
24	22, 23	bitri 252	. . . . . . . 8 |- ( y ( _V X. Singletons ) z <-> E. x z = { x } )
25	24	anbi2i 698	. . . . . . 7 |- ( ( y (Image F o. Singleton ) z /\ y ( _V X. Singletons ) z ) <-> ( y (Image F o. Singleton ) z /\ E. x z = { x } ) )
26		brin 4473	. . . . . . 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 1827	. . . . . . 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 280	. . . . . 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 1712	. . . . 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 1903	. . . . 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 252	. . . 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 1716	. . . . . 6 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. z ( z = { x } /\ y (Image F o. Singleton ) z ) )
33		snex 4662	. . . . . . 7 |- { x } e. _V
34		breq2 4427	. . . . . . 7 |- ( z = { x } -> ( y (Image F o. Singleton ) z <-> y (Image F o. Singleton ) { x } ) )
35	33, 34	ceqsexv 3118	. . . . . 6 |- ( E. z ( z = { x } /\ y (Image F o. Singleton ) z ) <-> y (Image F o. Singleton ) { x } )
36	19, 33	brco 5024	. . . . . . 7 |- ( y (Image F o. Singleton ) { x } <-> E. z ( ySingleton z /\ zImage F { x } ) )
37		vex 3083	. . . . . . . . . 10 |- z e. _V
38	19, 37	brsingle 30689	. . . . . . . . 9 |- ( ySingleton z <-> z = { y } )
39	38	anbi1i 699	. . . . . . . 8 |- ( ( ySingleton z /\ zImage F { x } ) <-> ( z = { y } /\ zImage F { x } ) )
40	39	exbii 1712	. . . . . . 7 |- ( E. z ( ySingleton z /\ zImage F { x } ) <-> E. z ( z = { y } /\ zImage F { x } ) )
41		snex 4662	. . . . . . . . 9 |- { y } e. _V
42		breq1 4426	. . . . . . . . 9 |- ( z = { y } -> ( zImage F { x } <-> { y }Image F { x } ) )
43	41, 42	ceqsexv 3118	. . . . . . . 8 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> { y }Image F { x } )
44	41, 33	brimage 30698	. . . . . . . 8 |- ( { y }Image F { x } <-> { x } = ( F " { y } ) )
45		eqcom 2431	. . . . . . . 8 |- ( { x } = ( F " { y } ) <-> ( F " { y } ) = { x } )
46	43, 44, 45	3bitri 274	. . . . . . 7 |- ( E. z ( z = { y } /\ zImage F { x } ) <-> ( F " { y } ) = { x } )
47	36, 40, 46	3bitri 274	. . . . . 6 |- ( y (Image F o. Singleton ) { x } <-> ( F " { y } ) = { x } )
48	32, 35, 47	3bitri 274	. . . . 5 |- ( E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> ( F " { y } ) = { x } )
49	48	exbii 1712	. . . 4 |- ( E. x E. z ( y (Image F o. Singleton ) z /\ z = { x } ) <-> E. x ( F " { y } ) = { x } )
50	20, 31, 49	3bitri 274	. . 3 |- ( y e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { y } ) = { x } )
51	14, 18, 50	vtoclbg 3140	. 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 354	1 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { A } ) = { x } )

Syntax hints:   -. wn 3   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   e. wcel 1872   _Vcvv 3080   i^i cin 3435   (/)c0 3761   {csn 3998   class class class wbr 4423   X. cxp 4851   dom cdm 4853   "cima 4856   o. ccom 4857  Singletoncsingle 30609   Singletonscsingles 30610  Imagecimage 30611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-symdif 3693  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-eprel 4764  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6807  df-2nd 6808  df-txp 30625  df-singleton 30633  df-singles 30634  df-image 30635

This theorem is referenced by:  funpartfun  30715  funpartfv  30717