Theorem ressress 15938

Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)

Assertion

Ref	Expression
ressress	|- ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )

Proof of Theorem ressress

Step	Hyp	Ref	Expression
1		simplr 792	. . . . . . . . 9 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> -. ( Base ` W ) C_ A )
2		simpr1 1067	. . . . . . . . 9 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> W e. _V )
3		simpr2 1068	. . . . . . . . 9 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> A e. X )
4		eqid 2622	. . . . . . . . . 10 |- ( W ↾s A ) = ( W ↾s A )
5		eqid 2622	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
6	4, 5	ressval2 15929	. . . . . . . . 9 |- ( ( -. ( Base ` W ) C_ A /\ W e. _V /\ A e. X ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
7	1, 2, 3, 6	syl3anc 1326	. . . . . . . 8 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
8		inass 3823	. . . . . . . . . . 11 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
9		in12 3824	. . . . . . . . . . 11 |- ( A i^i ( B i^i ( Base ` W ) ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
10	8, 9	eqtri 2644	. . . . . . . . . 10 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
11	4, 5	ressbas 15930	. . . . . . . . . . . 12 |- ( A e. X -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
12	3, 11	syl 17	. . . . . . . . . . 11 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
13	12	ineq2d 3814	. . . . . . . . . 10 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( A i^i ( Base ` W ) ) ) = ( B i^i ( Base ` ( W ↾s A ) ) ) )
14	10, 13	syl5req 2669	. . . . . . . . 9 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( Base ` ( W ↾s A ) ) ) = ( ( A i^i B ) i^i ( Base ` W ) ) )
15	14	opeq2d 4409	. . . . . . . 8 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. = <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. )
16	7, 15	oveq12d 6668	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) = ( ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
17		fvex 6201	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	17	inex2 4800	. . . . . . . 8 |- ( ( A i^i B ) i^i ( Base ` W ) ) e. _V
19		setsabs 15902	. . . . . . . 8 |- ( ( W e. _V /\ ( ( A i^i B ) i^i ( Base ` W ) ) e. _V ) -> ( ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
20	2, 18, 19	sylancl 694	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
21	16, 20	eqtrd 2656	. . . . . 6 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
22		simpll 790	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> -. ( Base ` ( W ↾s A ) ) C_ B )
23		ovexd 6680	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) e. _V )
24		simpr3 1069	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> B e. Y )
25		eqid 2622	. . . . . . . 8 |- ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) ↾s B )
26		eqid 2622	. . . . . . . 8 |- ( Base ` ( W ↾s A ) ) = ( Base ` ( W ↾s A ) )
27	25, 26	ressval2 15929	. . . . . . 7 |- ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ ( W ↾s A ) e. _V /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) )
28	22, 23, 24, 27	syl3anc 1326	. . . . . 6 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) )
29		inss1 3833	. . . . . . . . 9 |- ( A i^i B ) C_ A
30		sstr 3611	. . . . . . . . 9 |- ( ( ( Base ` W ) C_ ( A i^i B ) /\ ( A i^i B ) C_ A ) -> ( Base ` W ) C_ A )
31	29, 30	mpan2 707	. . . . . . . 8 |- ( ( Base ` W ) C_ ( A i^i B ) -> ( Base ` W ) C_ A )
32	1, 31	nsyl 135	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> -. ( Base ` W ) C_ ( A i^i B ) )
33		inex1g 4801	. . . . . . . 8 |- ( A e. X -> ( A i^i B ) e. _V )
34	3, 33	syl 17	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( A i^i B ) e. _V )
35		eqid 2622	. . . . . . . 8 |- ( W ↾s ( A i^i B ) ) = ( W ↾s ( A i^i B ) )
36	35, 5	ressval2 15929	. . . . . . 7 |- ( ( -. ( Base ` W ) C_ ( A i^i B ) /\ W e. _V /\ ( A i^i B ) e. _V ) -> ( W ↾s ( A i^i B ) ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
37	32, 2, 34, 36	syl3anc 1326	. . . . . 6 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s ( A i^i B ) ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
38	21, 28, 37	3eqtr4d 2666	. . . . 5 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
39	38	exp31 630	. . . 4 |- ( -. ( Base ` ( W ↾s A ) ) C_ B -> ( -. ( Base ` W ) C_ A -> ( ( W e. _V /\ A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) ) )
40		ovex 6678	. . . . . . . 8 |- ( W ↾s A ) e. _V
41	25, 26	ressid2 15928	. . . . . . . 8 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W ↾s A ) e. _V /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s A ) )
42	40, 41	mp3an2 1412	. . . . . . 7 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s A ) )
43	42	3ad2antr3 1228	. . . . . 6 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s A ) )
44		in32 3825	. . . . . . . . 9 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( ( A i^i ( Base ` W ) ) i^i B )
45		simpr2 1068	. . . . . . . . . . . 12 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> A e. X )
46	45, 11	syl 17	. . . . . . . . . . 11 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
47		simpl 473	. . . . . . . . . . 11 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( Base ` ( W ↾s A ) ) C_ B )
48	46, 47	eqsstrd 3639	. . . . . . . . . 10 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( A i^i ( Base ` W ) ) C_ B )
49		df-ss 3588	. . . . . . . . . 10 |- ( ( A i^i ( Base ` W ) ) C_ B <-> ( ( A i^i ( Base ` W ) ) i^i B ) = ( A i^i ( Base ` W ) ) )
50	48, 49	sylib 208	. . . . . . . . 9 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( A i^i ( Base ` W ) ) i^i B ) = ( A i^i ( Base ` W ) ) )
51	44, 50	syl5req 2669	. . . . . . . 8 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( A i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) ) )
52	51	oveq2d 6666	. . . . . . 7 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s ( A i^i ( Base ` W ) ) ) = ( W ↾s ( ( A i^i B ) i^i ( Base ` W ) ) ) )
53	5	ressinbas 15936	. . . . . . . 8 |- ( A e. X -> ( W ↾s A ) = ( W ↾s ( A i^i ( Base ` W ) ) ) )
54	45, 53	syl 17	. . . . . . 7 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = ( W ↾s ( A i^i ( Base ` W ) ) ) )
55	5	ressinbas 15936	. . . . . . . 8 |- ( ( A i^i B ) e. _V -> ( W ↾s ( A i^i B ) ) = ( W ↾s ( ( A i^i B ) i^i ( Base ` W ) ) ) )
56	45, 33, 55	3syl 18	. . . . . . 7 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s ( A i^i B ) ) = ( W ↾s ( ( A i^i B ) i^i ( Base ` W ) ) ) )
57	52, 54, 56	3eqtr4d 2666	. . . . . 6 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
58	43, 57	eqtrd 2656	. . . . 5 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
59	58	ex 450	. . . 4 |- ( ( Base ` ( W ↾s A ) ) C_ B -> ( ( W e. _V /\ A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
60	4, 5	ressid2 15928	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ W e. _V /\ A e. X ) -> ( W ↾s A ) = W )
61	60	3adant3r3 1276	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = W )
62	61	oveq1d 6665	. . . . . 6 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s B ) )
63		inss2 3834	. . . . . . . . . . 11 |- ( B i^i ( Base ` W ) ) C_ ( Base ` W )
64		simpl 473	. . . . . . . . . . 11 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( Base ` W ) C_ A )
65	63, 64	syl5ss 3614	. . . . . . . . . 10 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( Base ` W ) ) C_ A )
66		sseqin2 3817	. . . . . . . . . 10 |- ( ( B i^i ( Base ` W ) ) C_ A <-> ( A i^i ( B i^i ( Base ` W ) ) ) = ( B i^i ( Base ` W ) ) )
67	65, 66	sylib 208	. . . . . . . . 9 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( A i^i ( B i^i ( Base ` W ) ) ) = ( B i^i ( Base ` W ) ) )
68	8, 67	syl5req 2669	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) ) )
69	68	oveq2d 6666	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s ( B i^i ( Base ` W ) ) ) = ( W ↾s ( ( A i^i B ) i^i ( Base ` W ) ) ) )
70		simpr3 1069	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> B e. Y )
71	5	ressinbas 15936	. . . . . . . 8 |- ( B e. Y -> ( W ↾s B ) = ( W ↾s ( B i^i ( Base ` W ) ) ) )
72	70, 71	syl 17	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s B ) = ( W ↾s ( B i^i ( Base ` W ) ) ) )
73		simpr2 1068	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> A e. X )
74	73, 33, 55	3syl 18	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s ( A i^i B ) ) = ( W ↾s ( ( A i^i B ) i^i ( Base ` W ) ) ) )
75	69, 72, 74	3eqtr4d 2666	. . . . . 6 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s B ) = ( W ↾s ( A i^i B ) ) )
76	62, 75	eqtrd 2656	. . . . 5 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
77	76	ex 450	. . . 4 |- ( ( Base ` W ) C_ A -> ( ( W e. _V /\ A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
78	39, 59, 77	pm2.61ii 177	. . 3 |- ( ( W e. _V /\ A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
79	78	3expib 1268	. 2 |- ( W e. _V -> ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
80		ress0 15934	. . . 4 |- ( (/) ↾s B ) = (/)
81		reldmress 15926	. . . . . 6 |- Rel dom ↾s
82	81	ovprc1 6684	. . . . 5 |- ( -. W e. _V -> ( W ↾s A ) = (/) )
83	82	oveq1d 6665	. . . 4 |- ( -. W e. _V -> ( ( W ↾s A ) ↾s B ) = ( (/) ↾s B ) )
84	81	ovprc1 6684	. . . 4 |- ( -. W e. _V -> ( W ↾s ( A i^i B ) ) = (/) )
85	80, 83, 84	3eqtr4a 2682	. . 3 |- ( -. W e. _V -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
86	85	a1d 25	. 2 |- ( -. W e. _V -> ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
87	79, 86	pm2.61i 176	1 |- ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858

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  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-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865

This theorem is referenced by:  ressabs  15939  xrge00  29686  xrge0slmod  29844  esumpfinvallem  30136  lmhmlnmsplit  37657