Theorem mreacs 16319

Description: Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
mreacs	|- ( X e. V -> (ACS ` X ) e. (Moore ` ~P X ) )

Proof of Theorem mreacs

Dummy variables a b c x d e f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( x = X -> (ACS ` x ) = (ACS ` X ) )
2		pweq 4161	. . . 4 |- ( x = X -> ~P x = ~P X )
3	2	fveq2d 6195	. . 3 |- ( x = X -> (Moore ` ~P x ) = (Moore ` ~P X ) )
4	1, 3	eleq12d 2695	. 2 |- ( x = X -> ( (ACS ` x ) e. (Moore ` ~P x ) <-> (ACS ` X ) e. (Moore ` ~P X ) ) )
5		acsmre 16313	. . . . . . . 8 |- ( a e. (ACS ` x ) -> a e. (Moore ` x ) )
6		mresspw 16252	. . . . . . . 8 |- ( a e. (Moore ` x ) -> a C_ ~P x )
7	5, 6	syl 17	. . . . . . 7 |- ( a e. (ACS ` x ) -> a C_ ~P x )
8		selpw 4165	. . . . . . 7 |- ( a e. ~P ~P x <-> a C_ ~P x )
9	7, 8	sylibr 224	. . . . . 6 |- ( a e. (ACS ` x ) -> a e. ~P ~P x )
10	9	ssriv 3607	. . . . 5 |- (ACS ` x ) C_ ~P ~P x
11	10	a1i 11	. . . 4 |- ( T. -> (ACS ` x ) C_ ~P ~P x )
12		vex 3203	. . . . . . . 8 |- x e. _V
13		mremre 16264	. . . . . . . 8 |- ( x e. _V -> (Moore ` x ) e. (Moore ` ~P x ) )
14	12, 13	mp1i 13	. . . . . . 7 |- ( a C_ (ACS ` x ) -> (Moore ` x ) e. (Moore ` ~P x ) )
15	5	ssriv 3607	. . . . . . . 8 |- (ACS ` x ) C_ (Moore ` x )
16		sstr 3611	. . . . . . . 8 |- ( ( a C_ (ACS ` x ) /\ (ACS ` x ) C_ (Moore ` x ) ) -> a C_ (Moore ` x ) )
17	15, 16	mpan2 707	. . . . . . 7 |- ( a C_ (ACS ` x ) -> a C_ (Moore ` x ) )
18		mrerintcl 16257	. . . . . . 7 |- ( ( (Moore ` x ) e. (Moore ` ~P x ) /\ a C_ (Moore ` x ) ) -> ( ~P x i^i |^| a ) e. (Moore ` x ) )
19	14, 17, 18	syl2anc 693	. . . . . 6 |- ( a C_ (ACS ` x ) -> ( ~P x i^i |^| a ) e. (Moore ` x ) )
20		ssel2 3598	. . . . . . . . . . . . . . . 16 |- ( ( a C_ (ACS ` x ) /\ d e. a ) -> d e. (ACS ` x ) )
21	20	acsmred 16317	. . . . . . . . . . . . . . 15 |- ( ( a C_ (ACS ` x ) /\ d e. a ) -> d e. (Moore ` x ) )
22		eqid 2622	. . . . . . . . . . . . . . 15 |- (mrCls ` d ) = (mrCls ` d )
23	21, 22	mrcssvd 16283	. . . . . . . . . . . . . 14 |- ( ( a C_ (ACS ` x ) /\ d e. a ) -> ( (mrCls ` d ) ` c ) C_ x )
24	23	ralrimiva 2966	. . . . . . . . . . . . 13 |- ( a C_ (ACS ` x ) -> A. d e. a ( (mrCls ` d ) ` c ) C_ x )
25	24	adantr 481	. . . . . . . . . . . 12 |- ( ( a C_ (ACS ` x ) /\ c e. ~P x ) -> A. d e. a ( (mrCls ` d ) ` c ) C_ x )
26		iunss 4561	. . . . . . . . . . . 12 |- ( U_ d e. a ( (mrCls ` d ) ` c ) C_ x <-> A. d e. a ( (mrCls ` d ) ` c ) C_ x )
27	25, 26	sylibr 224	. . . . . . . . . . 11 |- ( ( a C_ (ACS ` x ) /\ c e. ~P x ) -> U_ d e. a ( (mrCls ` d ) ` c ) C_ x )
28	12	elpw2 4828	. . . . . . . . . . 11 |- ( U_ d e. a ( (mrCls ` d ) ` c ) e. ~P x <-> U_ d e. a ( (mrCls ` d ) ` c ) C_ x )
29	27, 28	sylibr 224	. . . . . . . . . 10 |- ( ( a C_ (ACS ` x ) /\ c e. ~P x ) -> U_ d e. a ( (mrCls ` d ) ` c ) e. ~P x )
30		eqid 2622	. . . . . . . . . 10 |- ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) )
31	29, 30	fmptd 6385	. . . . . . . . 9 |- ( a C_ (ACS ` x ) -> ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) : ~P x --> ~P x )
32		fssxp 6060	. . . . . . . . 9 |- ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) : ~P x --> ~P x -> ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) C_ ( ~P x X. ~P x ) )
33	31, 32	syl 17	. . . . . . . 8 |- ( a C_ (ACS ` x ) -> ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) C_ ( ~P x X. ~P x ) )
34		vpwex 4849	. . . . . . . . 9 |- ~P x e. _V
35	34, 34	xpex 6962	. . . . . . . 8 |- ( ~P x X. ~P x ) e. _V
36		ssexg 4804	. . . . . . . 8 |- ( ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) C_ ( ~P x X. ~P x ) /\ ( ~P x X. ~P x ) e. _V ) -> ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) e. _V )
37	33, 35, 36	sylancl 694	. . . . . . 7 |- ( a C_ (ACS ` x ) -> ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) e. _V )
38	20	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ d e. a ) -> d e. (ACS ` x ) )
39		elpwi 4168	. . . . . . . . . . . . . 14 |- ( b e. ~P x -> b C_ x )
40	39	ad2antlr 763	. . . . . . . . . . . . 13 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ d e. a ) -> b C_ x )
41	22	acsfiel2 16316	. . . . . . . . . . . . 13 |- ( ( d e. (ACS ` x ) /\ b C_ x ) -> ( b e. d <-> A. e e. ( ~P b i^i Fin ) ( (mrCls ` d ) ` e ) C_ b ) )
42	38, 40, 41	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ d e. a ) -> ( b e. d <-> A. e e. ( ~P b i^i Fin ) ( (mrCls ` d ) ` e ) C_ b ) )
43	42	ralbidva 2985	. . . . . . . . . . 11 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( A. d e. a b e. d <-> A. d e. a A. e e. ( ~P b i^i Fin ) ( (mrCls ` d ) ` e ) C_ b ) )
44		iunss 4561	. . . . . . . . . . . . 13 |- ( U_ d e. a ( (mrCls ` d ) ` e ) C_ b <-> A. d e. a ( (mrCls ` d ) ` e ) C_ b )
45	44	ralbii 2980	. . . . . . . . . . . 12 |- ( A. e e. ( ~P b i^i Fin ) U_ d e. a ( (mrCls ` d ) ` e ) C_ b <-> A. e e. ( ~P b i^i Fin ) A. d e. a ( (mrCls ` d ) ` e ) C_ b )
46		ralcom 3098	. . . . . . . . . . . 12 |- ( A. e e. ( ~P b i^i Fin ) A. d e. a ( (mrCls ` d ) ` e ) C_ b <-> A. d e. a A. e e. ( ~P b i^i Fin ) ( (mrCls ` d ) ` e ) C_ b )
47	45, 46	bitri 264	. . . . . . . . . . 11 |- ( A. e e. ( ~P b i^i Fin ) U_ d e. a ( (mrCls ` d ) ` e ) C_ b <-> A. d e. a A. e e. ( ~P b i^i Fin ) ( (mrCls ` d ) ` e ) C_ b )
48	43, 47	syl6bbr 278	. . . . . . . . . 10 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( A. d e. a b e. d <-> A. e e. ( ~P b i^i Fin ) U_ d e. a ( (mrCls ` d ) ` e ) C_ b ) )
49		elrint2 4519	. . . . . . . . . . 11 |- ( b e. ~P x -> ( b e. ( ~P x i^i |^| a ) <-> A. d e. a b e. d ) )
50	49	adantl 482	. . . . . . . . . 10 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( b e. ( ~P x i^i |^| a ) <-> A. d e. a b e. d ) )
51		funmpt 5926	. . . . . . . . . . . . 13 |- Fun ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) )
52		funiunfv 6506	. . . . . . . . . . . . 13 |- ( Fun ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> U_ e e. ( ~P b i^i Fin ) ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) = U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) )
53	51, 52	ax-mp 5	. . . . . . . . . . . 12 |- U_ e e. ( ~P b i^i Fin ) ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) = U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) )
54	53	sseq1i 3629	. . . . . . . . . . 11 |- ( U_ e e. ( ~P b i^i Fin ) ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) C_ b <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b )
55		iunss 4561	. . . . . . . . . . . 12 |- ( U_ e e. ( ~P b i^i Fin ) ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) C_ b <-> A. e e. ( ~P b i^i Fin ) ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) C_ b )
56		inss1 3833	. . . . . . . . . . . . . . . . 17 |- ( ~P b i^i Fin ) C_ ~P b
57		sspwb 4917	. . . . . . . . . . . . . . . . . . 19 |- ( b C_ x <-> ~P b C_ ~P x )
58	39, 57	sylib 208	. . . . . . . . . . . . . . . . . 18 |- ( b e. ~P x -> ~P b C_ ~P x )
59	58	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ~P b C_ ~P x )
60	56, 59	syl5ss 3614	. . . . . . . . . . . . . . . 16 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( ~P b i^i Fin ) C_ ~P x )
61	60	sselda 3603	. . . . . . . . . . . . . . 15 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ e e. ( ~P b i^i Fin ) ) -> e e. ~P x )
62	21, 22	mrcssvd 16283	. . . . . . . . . . . . . . . . . . 19 |- ( ( a C_ (ACS ` x ) /\ d e. a ) -> ( (mrCls ` d ) ` e ) C_ x )
63	62	ralrimiva 2966	. . . . . . . . . . . . . . . . . 18 |- ( a C_ (ACS ` x ) -> A. d e. a ( (mrCls ` d ) ` e ) C_ x )
64	63	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ e e. ( ~P b i^i Fin ) ) -> A. d e. a ( (mrCls ` d ) ` e ) C_ x )
65		iunss 4561	. . . . . . . . . . . . . . . . 17 |- ( U_ d e. a ( (mrCls ` d ) ` e ) C_ x <-> A. d e. a ( (mrCls ` d ) ` e ) C_ x )
66	64, 65	sylibr 224	. . . . . . . . . . . . . . . 16 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ e e. ( ~P b i^i Fin ) ) -> U_ d e. a ( (mrCls ` d ) ` e ) C_ x )
67		ssexg 4804	. . . . . . . . . . . . . . . 16 |- ( ( U_ d e. a ( (mrCls ` d ) ` e ) C_ x /\ x e. _V ) -> U_ d e. a ( (mrCls ` d ) ` e ) e. _V )
68	66, 12, 67	sylancl 694	. . . . . . . . . . . . . . 15 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ e e. ( ~P b i^i Fin ) ) -> U_ d e. a ( (mrCls ` d ) ` e ) e. _V )
69		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( c = e -> ( (mrCls ` d ) ` c ) = ( (mrCls ` d ) ` e ) )
70	69	iuneq2d 4547	. . . . . . . . . . . . . . . 16 |- ( c = e -> U_ d e. a ( (mrCls ` d ) ` c ) = U_ d e. a ( (mrCls ` d ) ` e ) )
71	70, 30	fvmptg 6280	. . . . . . . . . . . . . . 15 |- ( ( e e. ~P x /\ U_ d e. a ( (mrCls ` d ) ` e ) e. _V ) -> ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) = U_ d e. a ( (mrCls ` d ) ` e ) )
72	61, 68, 71	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ e e. ( ~P b i^i Fin ) ) -> ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) = U_ d e. a ( (mrCls ` d ) ` e ) )
73	72	sseq1d 3632	. . . . . . . . . . . . 13 |- ( ( ( a C_ (ACS ` x ) /\ b e. ~P x ) /\ e e. ( ~P b i^i Fin ) ) -> ( ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) C_ b <-> U_ d e. a ( (mrCls ` d ) ` e ) C_ b ) )
74	73	ralbidva 2985	. . . . . . . . . . . 12 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( A. e e. ( ~P b i^i Fin ) ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) C_ b <-> A. e e. ( ~P b i^i Fin ) U_ d e. a ( (mrCls ` d ) ` e ) C_ b ) )
75	55, 74	syl5bb 272	. . . . . . . . . . 11 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( U_ e e. ( ~P b i^i Fin ) ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) ` e ) C_ b <-> A. e e. ( ~P b i^i Fin ) U_ d e. a ( (mrCls ` d ) ` e ) C_ b ) )
76	54, 75	syl5bbr 274	. . . . . . . . . 10 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b <-> A. e e. ( ~P b i^i Fin ) U_ d e. a ( (mrCls ` d ) ` e ) C_ b ) )
77	48, 50, 76	3bitr4d 300	. . . . . . . . 9 |- ( ( a C_ (ACS ` x ) /\ b e. ~P x ) -> ( b e. ( ~P x i^i |^| a ) <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) )
78	77	ralrimiva 2966	. . . . . . . 8 |- ( a C_ (ACS ` x ) -> A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) )
79	31, 78	jca 554	. . . . . . 7 |- ( a C_ (ACS ` x ) -> ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) : ~P x --> ~P x /\ A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) ) )
80		feq1 6026	. . . . . . . . 9 |- ( f = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> ( f : ~P x --> ~P x <-> ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) : ~P x --> ~P x ) )
81		imaeq1 5461	. . . . . . . . . . . . 13 |- ( f = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> ( f " ( ~P b i^i Fin ) ) = ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) )
82	81	unieqd 4446	. . . . . . . . . . . 12 |- ( f = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> U. ( f " ( ~P b i^i Fin ) ) = U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) )
83	82	sseq1d 3632	. . . . . . . . . . 11 |- ( f = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> ( U. ( f " ( ~P b i^i Fin ) ) C_ b <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) )
84	83	bibi2d 332	. . . . . . . . . 10 |- ( f = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> ( ( b e. ( ~P x i^i |^| a ) <-> U. ( f " ( ~P b i^i Fin ) ) C_ b ) <-> ( b e. ( ~P x i^i |^| a ) <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) ) )
85	84	ralbidv 2986	. . . . . . . . 9 |- ( f = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> ( A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( f " ( ~P b i^i Fin ) ) C_ b ) <-> A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) ) )
86	80, 85	anbi12d 747	. . . . . . . 8 |- ( f = ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) -> ( ( f : ~P x --> ~P x /\ A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( f " ( ~P b i^i Fin ) ) C_ b ) ) <-> ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) : ~P x --> ~P x /\ A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) ) ) )
87	86	spcegv 3294	. . . . . . 7 |- ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) e. _V -> ( ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) : ~P x --> ~P x /\ A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( ( c e. ~P x |-> U_ d e. a ( (mrCls ` d ) ` c ) ) " ( ~P b i^i Fin ) ) C_ b ) ) -> E. f ( f : ~P x --> ~P x /\ A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( f " ( ~P b i^i Fin ) ) C_ b ) ) ) )
88	37, 79, 87	sylc 65	. . . . . 6 |- ( a C_ (ACS ` x ) -> E. f ( f : ~P x --> ~P x /\ A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( f " ( ~P b i^i Fin ) ) C_ b ) ) )
89		isacs 16312	. . . . . 6 |- ( ( ~P x i^i |^| a ) e. (ACS ` x ) <-> ( ( ~P x i^i |^| a ) e. (Moore ` x ) /\ E. f ( f : ~P x --> ~P x /\ A. b e. ~P x ( b e. ( ~P x i^i |^| a ) <-> U. ( f " ( ~P b i^i Fin ) ) C_ b ) ) ) )
90	19, 88, 89	sylanbrc 698	. . . . 5 |- ( a C_ (ACS ` x ) -> ( ~P x i^i |^| a ) e. (ACS ` x ) )
91	90	adantl 482	. . . 4 |- ( ( T. /\ a C_ (ACS ` x ) ) -> ( ~P x i^i |^| a ) e. (ACS ` x ) )
92	11, 91	ismred2 16263	. . 3 |- ( T. -> (ACS ` x ) e. (Moore ` ~P x ) )
93	92	trud 1493	. 2 |- (ACS ` x ) e. (Moore ` ~P x )
94	4, 93	vtoclg 3266	1 |- ( X e. V -> (ACS ` X ) e. (Moore ` ~P X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888   Fincfn 7955  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245

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-iun 4522  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  df-acs 16249

This theorem is referenced by:  acsfn1  16322  acsfn1c  16323  acsfn2  16324  submacs  17365  subgacs  17629  nsgacs  17630  lssacs  18967  acsfn1p  37769  subrgacs  37770  sdrgacs  37771