Theorem mrcval 16270

Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Hypothesis

Ref	Expression
mrcfval.f	|- F = (mrCls ` C )

Assertion

Ref	Expression
mrcval	|- ( ( C e. (Moore ` X ) /\ U C_ X ) -> ( F ` U ) = |^| { s e. C | U C_ s } )

Distinct variable groups:   F, s   C, s   X, s   U, s

Proof of Theorem mrcval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrcfval.f	. . . 4 |- F = (mrCls ` C )
2	1	mrcfval 16268	. . 3 |- ( C e. (Moore ` X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )
3	2	adantr 481	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )
4		sseq1 3626	. . . . 5 |- ( x = U -> ( x C_ s <-> U C_ s ) )
5	4	rabbidv 3189	. . . 4 |- ( x = U -> { s e. C | x C_ s } = { s e. C | U C_ s } )
6	5	inteqd 4480	. . 3 |- ( x = U -> |^| { s e. C | x C_ s } = |^| { s e. C | U C_ s } )
7	6	adantl 482	. 2 |- ( ( ( C e. (Moore ` X ) /\ U C_ X ) /\ x = U ) -> |^| { s e. C | x C_ s } = |^| { s e. C | U C_ s } )
8		mre1cl 16254	. . . 4 |- ( C e. (Moore ` X ) -> X e. C )
9		elpw2g 4827	. . . 4 |- ( X e. C -> ( U e. ~P X <-> U C_ X ) )
10	8, 9	syl 17	. . 3 |- ( C e. (Moore ` X ) -> ( U e. ~P X <-> U C_ X ) )
11	10	biimpar 502	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> U e. ~P X )
12	8	adantr 481	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> X e. C )
13		simpr 477	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> U C_ X )
14		sseq2 3627	. . . . . 6 |- ( s = X -> ( U C_ s <-> U C_ X ) )
15	14	elrab 3363	. . . . 5 |- ( X e. { s e. C | U C_ s } <-> ( X e. C /\ U C_ X ) )
16	12, 13, 15	sylanbrc 698	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> X e. { s e. C | U C_ s } )
17		ne0i 3921	. . . 4 |- ( X e. { s e. C | U C_ s } -> { s e. C | U C_ s } =/= (/) )
18	16, 17	syl 17	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> { s e. C | U C_ s } =/= (/) )
19		intex 4820	. . 3 |- ( { s e. C | U C_ s } =/= (/) <-> |^| { s e. C | U C_ s } e. _V )
20	18, 19	sylib 208	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> |^| { s e. C | U C_ s } e. _V )
21	3, 7, 11, 20	fvmptd 6288	1 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> ( F ` U ) = |^| { s e. C | U C_ s } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243

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  ax-un 6949

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-csb 3534  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-int 4476  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246  df-mrc 16247

This theorem is referenced by:  mrcid  16273  mrcss  16276  mrcssid  16277  cycsubg2  17631  aspval2  19347