Theorem mrcuni 16281

Description: Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

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

Assertion

Ref	Expression
mrcuni	|- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` U. U ) = ( F ` U. ( F " U ) ) )

Proof of Theorem mrcuni

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

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> C e. (Moore ` X ) )
2		simpll 790	. . . . . . 7 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ s e. U ) -> C e. (Moore ` X ) )
3		ssel2 3598	. . . . . . . . 9 |- ( ( U C_ ~P X /\ s e. U ) -> s e. ~P X )
4	3	elpwid 4170	. . . . . . . 8 |- ( ( U C_ ~P X /\ s e. U ) -> s C_ X )
5	4	adantll 750	. . . . . . 7 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ s e. U ) -> s C_ X )
6		mrcfval.f	. . . . . . . 8 |- F = (mrCls ` C )
7	6	mrcssid 16277	. . . . . . 7 |- ( ( C e. (Moore ` X ) /\ s C_ X ) -> s C_ ( F ` s ) )
8	2, 5, 7	syl2anc 693	. . . . . 6 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ s e. U ) -> s C_ ( F ` s ) )
9	6	mrcf 16269	. . . . . . . . . . 11 |- ( C e. (Moore ` X ) -> F : ~P X --> C )
10		ffun 6048	. . . . . . . . . . 11 |- ( F : ~P X --> C -> Fun F )
11	9, 10	syl 17	. . . . . . . . . 10 |- ( C e. (Moore ` X ) -> Fun F )
12	11	adantr 481	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> Fun F )
13		fdm 6051	. . . . . . . . . . . 12 |- ( F : ~P X --> C -> dom F = ~P X )
14	9, 13	syl 17	. . . . . . . . . . 11 |- ( C e. (Moore ` X ) -> dom F = ~P X )
15	14	sseq2d 3633	. . . . . . . . . 10 |- ( C e. (Moore ` X ) -> ( U C_ dom F <-> U C_ ~P X ) )
16	15	biimpar 502	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> U C_ dom F )
17		funfvima2 6493	. . . . . . . . 9 |- ( ( Fun F /\ U C_ dom F ) -> ( s e. U -> ( F ` s ) e. ( F " U ) ) )
18	12, 16, 17	syl2anc 693	. . . . . . . 8 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( s e. U -> ( F ` s ) e. ( F " U ) ) )
19	18	imp 445	. . . . . . 7 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ s e. U ) -> ( F ` s ) e. ( F " U ) )
20		elssuni 4467	. . . . . . 7 |- ( ( F ` s ) e. ( F " U ) -> ( F ` s ) C_ U. ( F " U ) )
21	19, 20	syl 17	. . . . . 6 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ s e. U ) -> ( F ` s ) C_ U. ( F " U ) )
22	8, 21	sstrd 3613	. . . . 5 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ s e. U ) -> s C_ U. ( F " U ) )
23	22	ralrimiva 2966	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> A. s e. U s C_ U. ( F " U ) )
24		unissb 4469	. . . 4 |- ( U. U C_ U. ( F " U ) <-> A. s e. U s C_ U. ( F " U ) )
25	23, 24	sylibr 224	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> U. U C_ U. ( F " U ) )
26	6	mrcssv 16274	. . . . . . 7 |- ( C e. (Moore ` X ) -> ( F ` x ) C_ X )
27	26	adantr 481	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` x ) C_ X )
28	27	ralrimivw 2967	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> A. x e. U ( F ` x ) C_ X )
29		ffn 6045	. . . . . . 7 |- ( F : ~P X --> C -> F Fn ~P X )
30	9, 29	syl 17	. . . . . 6 |- ( C e. (Moore ` X ) -> F Fn ~P X )
31		sseq1 3626	. . . . . . 7 |- ( s = ( F ` x ) -> ( s C_ X <-> ( F ` x ) C_ X ) )
32	31	ralima 6498	. . . . . 6 |- ( ( F Fn ~P X /\ U C_ ~P X ) -> ( A. s e. ( F " U ) s C_ X <-> A. x e. U ( F ` x ) C_ X ) )
33	30, 32	sylan 488	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( A. s e. ( F " U ) s C_ X <-> A. x e. U ( F ` x ) C_ X ) )
34	28, 33	mpbird 247	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> A. s e. ( F " U ) s C_ X )
35		unissb 4469	. . . 4 |- ( U. ( F " U ) C_ X <-> A. s e. ( F " U ) s C_ X )
36	34, 35	sylibr 224	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> U. ( F " U ) C_ X )
37	6	mrcss 16276	. . 3 |- ( ( C e. (Moore ` X ) /\ U. U C_ U. ( F " U ) /\ U. ( F " U ) C_ X ) -> ( F ` U. U ) C_ ( F ` U. ( F " U ) ) )
38	1, 25, 36, 37	syl3anc 1326	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` U. U ) C_ ( F ` U. ( F " U ) ) )
39		simpll 790	. . . . . . . 8 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ x e. U ) -> C e. (Moore ` X ) )
40		elssuni 4467	. . . . . . . . 9 |- ( x e. U -> x C_ U. U )
41	40	adantl 482	. . . . . . . 8 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ x e. U ) -> x C_ U. U )
42		sspwuni 4611	. . . . . . . . . . 11 |- ( U C_ ~P X <-> U. U C_ X )
43	42	biimpi 206	. . . . . . . . . 10 |- ( U C_ ~P X -> U. U C_ X )
44	43	adantl 482	. . . . . . . . 9 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> U. U C_ X )
45	44	adantr 481	. . . . . . . 8 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ x e. U ) -> U. U C_ X )
46	6	mrcss 16276	. . . . . . . 8 |- ( ( C e. (Moore ` X ) /\ x C_ U. U /\ U. U C_ X ) -> ( F ` x ) C_ ( F ` U. U ) )
47	39, 41, 45, 46	syl3anc 1326	. . . . . . 7 |- ( ( ( C e. (Moore ` X ) /\ U C_ ~P X ) /\ x e. U ) -> ( F ` x ) C_ ( F ` U. U ) )
48	47	ralrimiva 2966	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> A. x e. U ( F ` x ) C_ ( F ` U. U ) )
49		sseq1 3626	. . . . . . . 8 |- ( s = ( F ` x ) -> ( s C_ ( F ` U. U ) <-> ( F ` x ) C_ ( F ` U. U ) ) )
50	49	ralima 6498	. . . . . . 7 |- ( ( F Fn ~P X /\ U C_ ~P X ) -> ( A. s e. ( F " U ) s C_ ( F ` U. U ) <-> A. x e. U ( F ` x ) C_ ( F ` U. U ) ) )
51	30, 50	sylan 488	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( A. s e. ( F " U ) s C_ ( F ` U. U ) <-> A. x e. U ( F ` x ) C_ ( F ` U. U ) ) )
52	48, 51	mpbird 247	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> A. s e. ( F " U ) s C_ ( F ` U. U ) )
53		unissb 4469	. . . . 5 |- ( U. ( F " U ) C_ ( F ` U. U ) <-> A. s e. ( F " U ) s C_ ( F ` U. U ) )
54	52, 53	sylibr 224	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> U. ( F " U ) C_ ( F ` U. U ) )
55	6	mrcssv 16274	. . . . 5 |- ( C e. (Moore ` X ) -> ( F ` U. U ) C_ X )
56	55	adantr 481	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` U. U ) C_ X )
57	6	mrcss 16276	. . . 4 |- ( ( C e. (Moore ` X ) /\ U. ( F " U ) C_ ( F ` U. U ) /\ ( F ` U. U ) C_ X ) -> ( F ` U. ( F " U ) ) C_ ( F ` ( F ` U. U ) ) )
58	1, 54, 56, 57	syl3anc 1326	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` U. ( F " U ) ) C_ ( F ` ( F ` U. U ) ) )
59	6	mrcidm 16279	. . . 4 |- ( ( C e. (Moore ` X ) /\ U. U C_ X ) -> ( F ` ( F ` U. U ) ) = ( F ` U. U ) )
60	1, 44, 59	syl2anc 693	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` ( F ` U. U ) ) = ( F ` U. U ) )
61	58, 60	sseqtrd 3641	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` U. ( F " U ) ) C_ ( F ` U. U ) )
62	38, 61	eqssd 3620	1 |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` U. U ) = ( F ` U. ( F " U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   dom cdm 5114   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` 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:  mrcun  16282  isacs4lem  17168