Theorem ismre 16250

Description: Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
ismre	|- ( C e. (Moore ` X ) <-> ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) )

Distinct variable groups:   C, s   X, s

Proof of Theorem ismre

Dummy variables c x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. 2 |- ( C e. (Moore ` X ) -> X e. _V )
2		elex 3212	. . 3 |- ( X e. C -> X e. _V )
3	2	3ad2ant2 1083	. 2 |- ( ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) -> X e. _V )
4		pweq 4161	. . . . . . 7 |- ( x = X -> ~P x = ~P X )
5	4	pweqd 4163	. . . . . 6 |- ( x = X -> ~P ~P x = ~P ~P X )
6		eleq1 2689	. . . . . . 7 |- ( x = X -> ( x e. c <-> X e. c ) )
7	6	anbi1d 741	. . . . . 6 |- ( x = X -> ( ( x e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) <-> ( X e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) ) )
8	5, 7	rabeqbidv 3195	. . . . 5 |- ( x = X -> { c e. ~P ~P x | ( x e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } = { c e. ~P ~P X | ( X e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } )
9		df-mre 16246	. . . . 5 |- Moore = ( x e. _V |-> { c e. ~P ~P x | ( x e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } )
10		vpwex 4849	. . . . . . 7 |- ~P x e. _V
11	10	pwex 4848	. . . . . 6 |- ~P ~P x e. _V
12	11	rabex 4813	. . . . 5 |- { c e. ~P ~P x | ( x e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } e. _V
13	8, 9, 12	fvmpt3i 6287	. . . 4 |- ( X e. _V -> (Moore ` X ) = { c e. ~P ~P X | ( X e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } )
14	13	eleq2d 2687	. . 3 |- ( X e. _V -> ( C e. (Moore ` X ) <-> C e. { c e. ~P ~P X | ( X e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } ) )
15		eleq2 2690	. . . . . 6 |- ( c = C -> ( X e. c <-> X e. C ) )
16		pweq 4161	. . . . . . 7 |- ( c = C -> ~P c = ~P C )
17		eleq2 2690	. . . . . . . 8 |- ( c = C -> ( |^| s e. c <-> |^| s e. C ) )
18	17	imbi2d 330	. . . . . . 7 |- ( c = C -> ( ( s =/= (/) -> |^| s e. c ) <-> ( s =/= (/) -> |^| s e. C ) ) )
19	16, 18	raleqbidv 3152	. . . . . 6 |- ( c = C -> ( A. s e. ~P c ( s =/= (/) -> |^| s e. c ) <-> A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) )
20	15, 19	anbi12d 747	. . . . 5 |- ( c = C -> ( ( X e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) <-> ( X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) )
21	20	elrab 3363	. . . 4 |- ( C e. { c e. ~P ~P X | ( X e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } <-> ( C e. ~P ~P X /\ ( X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) )
22	21	a1i 11	. . 3 |- ( X e. _V -> ( C e. { c e. ~P ~P X | ( X e. c /\ A. s e. ~P c ( s =/= (/) -> |^| s e. c ) ) } <-> ( C e. ~P ~P X /\ ( X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) ) )
23		pwexg 4850	. . . . . 6 |- ( X e. _V -> ~P X e. _V )
24		elpw2g 4827	. . . . . 6 |- ( ~P X e. _V -> ( C e. ~P ~P X <-> C C_ ~P X ) )
25	23, 24	syl 17	. . . . 5 |- ( X e. _V -> ( C e. ~P ~P X <-> C C_ ~P X ) )
26	25	anbi1d 741	. . . 4 |- ( X e. _V -> ( ( C e. ~P ~P X /\ ( X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) <-> ( C C_ ~P X /\ ( X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) ) )
27		3anass 1042	. . . 4 |- ( ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) <-> ( C C_ ~P X /\ ( X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) )
28	26, 27	syl6bbr 278	. . 3 |- ( X e. _V -> ( ( C e. ~P ~P X /\ ( X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) <-> ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) )
29	14, 22, 28	3bitrd 294	. 2 |- ( X e. _V -> ( C e. (Moore ` X ) <-> ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) ) )
30	1, 3, 29	pm5.21nii 368	1 |- ( C e. (Moore ` X ) <-> ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ` cfv 5888  Moorecmre 16242

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

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-nul 3916  df-if 4087  df-pw 4160  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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246

This theorem is referenced by:  mresspw  16252  mre1cl  16254  mreintcl  16255  ismred  16262