Theorem rescabs 16493

Description: Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
rescabs.c	|- ( ph -> C e. V )
rescabs.h	|- ( ph -> H Fn ( S X. S ) )
rescabs.j	|- ( ph -> J Fn ( T X. T ) )
rescabs.s	|- ( ph -> S e. W )
rescabs.t	|- ( ph -> T C_ S )

Assertion

Ref	Expression
rescabs	|- ( ph -> ( ( C |`cat H ) |`cat J ) = ( C |`cat J ) )

Proof of Theorem rescabs

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) |`cat J ) = ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) |`cat J )
2		ovexd 6680	. . . 4 |- ( ph -> ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) e. _V )
3		rescabs.s	. . . . 5 |- ( ph -> S e. W )
4		rescabs.t	. . . . 5 |- ( ph -> T C_ S )
5	3, 4	ssexd 4805	. . . 4 |- ( ph -> T e. _V )
6		rescabs.j	. . . 4 |- ( ph -> J Fn ( T X. T ) )
7	1, 2, 5, 6	rescval2 16488	. . 3 |- ( ph -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) |`cat J ) = ( ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
8		simpr 477	. . . . . . 7 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( Base ` ( C ↾s S ) ) C_ T )
9		ovexd 6680	. . . . . . 7 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) e. _V )
10	5	adantr 481	. . . . . . 7 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> T e. _V )
11		eqid 2622	. . . . . . . 8 |- ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) = ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T )
12		baseid 15919	. . . . . . . . 9 |- Base = Slot ( Base ` ndx )
13		1re 10039	. . . . . . . . . . 11 |- 1 e. RR
14		1nn 11031	. . . . . . . . . . . 12 |- 1 e. NN
15		4nn0 11311	. . . . . . . . . . . 12 |- 4 e. NN0
16		1nn0 11308	. . . . . . . . . . . 12 |- 1 e. NN0
17		1lt10 11681	. . . . . . . . . . . 12 |- 1 < ; 1 0
18	14, 15, 16, 17	declti 11546	. . . . . . . . . . 11 |- 1 < ; 1 4
19	13, 18	ltneii 10150	. . . . . . . . . 10 |- 1 =/= ; 1 4
20		basendx 15923	. . . . . . . . . . 11 |- ( Base ` ndx ) = 1
21		homndx 16074	. . . . . . . . . . 11 |- ( Hom ` ndx ) = ; 1 4
22	20, 21	neeq12i 2860	. . . . . . . . . 10 |- ( ( Base ` ndx ) =/= ( Hom ` ndx ) <-> 1 =/= ; 1 4 )
23	19, 22	mpbir 221	. . . . . . . . 9 |- ( Base ` ndx ) =/= ( Hom ` ndx )
24	12, 23	setsnid 15915	. . . . . . . 8 |- ( Base ` ( C ↾s S ) ) = ( Base ` ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) )
25	11, 24	ressid2 15928	. . . . . . 7 |- ( ( ( Base ` ( C ↾s S ) ) C_ T /\ ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) e. _V /\ T e. _V ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) = ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) )
26	8, 9, 10, 25	syl3anc 1326	. . . . . 6 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) = ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) )
27	26	oveq1d 6665	. . . . 5 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Hom ` ndx ) , J >. ) )
28		ovex 6678	. . . . . 6 |- ( C ↾s S ) e. _V
29		xpexg 6960	. . . . . . . . 9 |- ( ( T e. _V /\ T e. _V ) -> ( T X. T ) e. _V )
30	5, 5, 29	syl2anc 693	. . . . . . . 8 |- ( ph -> ( T X. T ) e. _V )
31		fnex 6481	. . . . . . . 8 |- ( ( J Fn ( T X. T ) /\ ( T X. T ) e. _V ) -> J e. _V )
32	6, 30, 31	syl2anc 693	. . . . . . 7 |- ( ph -> J e. _V )
33	32	adantr 481	. . . . . 6 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> J e. _V )
34		setsabs 15902	. . . . . 6 |- ( ( ( C ↾s S ) e. _V /\ J e. _V ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , J >. ) )
35	28, 33, 34	sylancr 695	. . . . 5 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , J >. ) )
36		eqid 2622	. . . . . . . . . . . . . 14 |- ( C ↾s S ) = ( C ↾s S )
37		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` C ) = ( Base ` C )
38	36, 37	ressbas 15930	. . . . . . . . . . . . 13 |- ( S e. W -> ( S i^i ( Base ` C ) ) = ( Base ` ( C ↾s S ) ) )
39	3, 38	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( S i^i ( Base ` C ) ) = ( Base ` ( C ↾s S ) ) )
40	39	sseq1d 3632	. . . . . . . . . . 11 |- ( ph -> ( ( S i^i ( Base ` C ) ) C_ T <-> ( Base ` ( C ↾s S ) ) C_ T ) )
41	40	biimpar 502	. . . . . . . . . 10 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( S i^i ( Base ` C ) ) C_ T )
42		inss2 3834	. . . . . . . . . . 11 |- ( S i^i ( Base ` C ) ) C_ ( Base ` C )
43	42	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( S i^i ( Base ` C ) ) C_ ( Base ` C ) )
44	41, 43	ssind 3837	. . . . . . . . 9 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( S i^i ( Base ` C ) ) C_ ( T i^i ( Base ` C ) ) )
45	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> T C_ S )
46		ssrin 3838	. . . . . . . . . 10 |- ( T C_ S -> ( T i^i ( Base ` C ) ) C_ ( S i^i ( Base ` C ) ) )
47	45, 46	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( T i^i ( Base ` C ) ) C_ ( S i^i ( Base ` C ) ) )
48	44, 47	eqssd 3620	. . . . . . . 8 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( S i^i ( Base ` C ) ) = ( T i^i ( Base ` C ) ) )
49	48	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( C ↾s ( S i^i ( Base ` C ) ) ) = ( C ↾s ( T i^i ( Base ` C ) ) ) )
50	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> S e. W )
51	37	ressinbas 15936	. . . . . . . 8 |- ( S e. W -> ( C ↾s S ) = ( C ↾s ( S i^i ( Base ` C ) ) ) )
52	50, 51	syl 17	. . . . . . 7 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( C ↾s S ) = ( C ↾s ( S i^i ( Base ` C ) ) ) )
53	37	ressinbas 15936	. . . . . . . 8 |- ( T e. _V -> ( C ↾s T ) = ( C ↾s ( T i^i ( Base ` C ) ) ) )
54	10, 53	syl 17	. . . . . . 7 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( C ↾s T ) = ( C ↾s ( T i^i ( Base ` C ) ) ) )
55	49, 52, 54	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( C ↾s S ) = ( C ↾s T ) )
56	55	oveq1d 6665	. . . . 5 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
57	27, 35, 56	3eqtrd 2660	. . . 4 |- ( ( ph /\ ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
58		simpr 477	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> -. ( Base ` ( C ↾s S ) ) C_ T )
59		ovexd 6680	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) e. _V )
60	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> T e. _V )
61	11, 24	ressval2 15929	. . . . . . . 8 |- ( ( -. ( Base ` ( C ↾s S ) ) C_ T /\ ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) e. _V /\ T e. _V ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) = ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) )
62	58, 59, 60, 61	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) = ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) )
63		ovexd 6680	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( C ↾s S ) e. _V )
64	23	necomi 2848	. . . . . . . . 9 |- ( Hom ` ndx ) =/= ( Base ` ndx )
65	64	a1i 11	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( Hom ` ndx ) =/= ( Base ` ndx ) )
66		rescabs.h	. . . . . . . . . 10 |- ( ph -> H Fn ( S X. S ) )
67		xpexg 6960	. . . . . . . . . . 11 |- ( ( S e. W /\ S e. W ) -> ( S X. S ) e. _V )
68	3, 3, 67	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( S X. S ) e. _V )
69		fnex 6481	. . . . . . . . . 10 |- ( ( H Fn ( S X. S ) /\ ( S X. S ) e. _V ) -> H e. _V )
70	66, 68, 69	syl2anc 693	. . . . . . . . 9 |- ( ph -> H e. _V )
71	70	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> H e. _V )
72		fvex 6201	. . . . . . . . . 10 |- ( Base ` ( C ↾s S ) ) e. _V
73	72	inex2 4800	. . . . . . . . 9 |- ( T i^i ( Base ` ( C ↾s S ) ) ) e. _V
74	73	a1i 11	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( T i^i ( Base ` ( C ↾s S ) ) ) e. _V )
75		fvex 6201	. . . . . . . . 9 |- ( Hom ` ndx ) e. _V
76		fvex 6201	. . . . . . . . 9 |- ( Base ` ndx ) e. _V
77	75, 76	setscom 15903	. . . . . . . 8 |- ( ( ( ( C ↾s S ) e. _V /\ ( Hom ` ndx ) =/= ( Base ` ndx ) ) /\ ( H e. _V /\ ( T i^i ( Base ` ( C ↾s S ) ) ) e. _V ) ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) = ( ( ( C ↾s S ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) sSet <. ( Hom ` ndx ) , H >. ) )
78	63, 65, 71, 74, 77	syl22anc 1327	. . . . . . 7 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) = ( ( ( C ↾s S ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) sSet <. ( Hom ` ndx ) , H >. ) )
79		eqid 2622	. . . . . . . . . . 11 |- ( ( C ↾s S ) ↾s T ) = ( ( C ↾s S ) ↾s T )
80		eqid 2622	. . . . . . . . . . 11 |- ( Base ` ( C ↾s S ) ) = ( Base ` ( C ↾s S ) )
81	79, 80	ressval2 15929	. . . . . . . . . 10 |- ( ( -. ( Base ` ( C ↾s S ) ) C_ T /\ ( C ↾s S ) e. _V /\ T e. _V ) -> ( ( C ↾s S ) ↾s T ) = ( ( C ↾s S ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) )
82	58, 63, 60, 81	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( C ↾s S ) ↾s T ) = ( ( C ↾s S ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) )
83	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> S e. W )
84	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> T C_ S )
85		ressabs 15939	. . . . . . . . . 10 |- ( ( S e. W /\ T C_ S ) -> ( ( C ↾s S ) ↾s T ) = ( C ↾s T ) )
86	83, 84, 85	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( C ↾s S ) ↾s T ) = ( C ↾s T ) )
87	82, 86	eqtr3d 2658	. . . . . . . 8 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( C ↾s S ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) = ( C ↾s T ) )
88	87	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( C ↾s S ) sSet <. ( Base ` ndx ) , ( T i^i ( Base ` ( C ↾s S ) ) ) >. ) sSet <. ( Hom ` ndx ) , H >. ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , H >. ) )
89	62, 78, 88	3eqtrd 2660	. . . . . 6 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , H >. ) )
90	89	oveq1d 6665	. . . . 5 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Hom ` ndx ) , J >. ) )
91		ovex 6678	. . . . . 6 |- ( C ↾s T ) e. _V
92	32	adantr 481	. . . . . 6 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> J e. _V )
93		setsabs 15902	. . . . . 6 |- ( ( ( C ↾s T ) e. _V /\ J e. _V ) -> ( ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
94	91, 92, 93	sylancr 695	. . . . 5 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , H >. ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
95	90, 94	eqtrd 2656	. . . 4 |- ( ( ph /\ -. ( Base ` ( C ↾s S ) ) C_ T ) -> ( ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
96	57, 95	pm2.61dan 832	. . 3 |- ( ph -> ( ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
97	7, 96	eqtrd 2656	. 2 |- ( ph -> ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) |`cat J ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
98		eqid 2622	. . . 4 |- ( C |`cat H ) = ( C |`cat H )
99		rescabs.c	. . . 4 |- ( ph -> C e. V )
100	98, 99, 3, 66	rescval2 16488	. . 3 |- ( ph -> ( C |`cat H ) = ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) )
101	100	oveq1d 6665	. 2 |- ( ph -> ( ( C |`cat H ) |`cat J ) = ( ( ( C ↾s S ) sSet <. ( Hom ` ndx ) , H >. ) |`cat J ) )
102		eqid 2622	. . 3 |- ( C |`cat J ) = ( C |`cat J )
103	102, 99, 5, 6	rescval2 16488	. 2 |- ( ph -> ( C |`cat J ) = ( ( C ↾s T ) sSet <. ( Hom ` ndx ) , J >. ) )
104	97, 101, 103	3eqtr4d 2666	1 |- ( ph -> ( ( C |`cat H ) |`cat J ) = ( C |`cat J ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1c1 9937   4c4 11072  ;cdc 11493   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾<sub>s</sub> cress 15858   Hom chom 15952   |`_cat cresc 16468

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-hom 15966  df-resc 16471

This theorem is referenced by:  subsubc  16513  fldc  42083  fldcALTV  42101