Theorem ressress 14548

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 754	. . . . . . . . 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 1002	. . . . . . . . 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 1003	. . . . . . . . 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 2467	. . . . . . . . . 10 |- ( W ↾s A ) = ( W ↾s A )
5		eqid 2467	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
6	4, 5	ressval2 14540	. . . . . . . . 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 1228	. . . . . . . 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 3708	. . . . . . . . . . 11 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
9		in12 3709	. . . . . . . . . . 11 |- ( A i^i ( B i^i ( Base ` W ) ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
10	8, 9	eqtri 2496	. . . . . . . . . 10 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
11	4, 5	ressbas 14541	. . . . . . . . . . . 12 |- ( A e. X -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
12	3, 11	syl 16	. . . . . . . . . . 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 3700	. . . . . . . . . 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 2521	. . . . . . . . 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 4220	. . . . . . . 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 6300	. . . . . . 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 5874	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	17	inex2 4589	. . . . . . . 8 |- ( ( A i^i B ) i^i ( Base ` W ) ) e. _V
19		setsabs 14515	. . . . . . . 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 662	. . . . . . 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 2508	. . . . . 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 753	. . . . . . 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		ovex 6307	. . . . . . . 8 |- ( W ↾s A ) e. _V
24	23	a1i 11	. . . . . . 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 )
25		simpr3 1004	. . . . . . 7 |- ( ( ( -. ( Base ` ( W ↾s A ) ) C_ B /\ -. ( Base ` W ) C_ A ) /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> B e. Y )
26		eqid 2467	. . . . . . . 8 |- ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) ↾s B )
27		eqid 2467	. . . . . . . 8 |- ( Base ` ( W ↾s A ) ) = ( Base ` ( W ↾s A ) )
28	26, 27	ressval2 14540	. . . . . . 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 ) ) ) >. ) )
29	22, 24, 25, 28	syl3anc 1228	. . . . . 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 ) ) ) >. ) )
30		inss1 3718	. . . . . . . . 9 |- ( A i^i B ) C_ A
31		sstr 3512	. . . . . . . . 9 |- ( ( ( Base ` W ) C_ ( A i^i B ) /\ ( A i^i B ) C_ A ) -> ( Base ` W ) C_ A )
32	30, 31	mpan2 671	. . . . . . . 8 |- ( ( Base ` W ) C_ ( A i^i B ) -> ( Base ` W ) C_ A )
33	1, 32	nsyl 121	. . . . . . 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 ) )
34		inex1g 4590	. . . . . . . 8 |- ( A e. X -> ( A i^i B ) e. _V )
35	3, 34	syl 16	. . . . . . 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 )
36		eqid 2467	. . . . . . . 8 |- ( W ↾s ( A i^i B ) ) = ( W ↾s ( A i^i B ) )
37	36, 5	ressval2 14540	. . . . . . 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 ) ) >. ) )
38	33, 2, 35, 37	syl3anc 1228	. . . . . 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 ) ) >. ) )
39	21, 29, 38	3eqtr4d 2518	. . . . 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 ) ) )
40	39	exp31 604	. . . 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 ) ) ) ) )
41	26, 27	ressid2 14539	. . . . . . . 8 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W ↾s A ) e. _V /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s A ) )
42	23, 41	mp3an2 1312	. . . . . . 7 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s A ) )
43	42	3ad2antr3 1163	. . . . . 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 3710	. . . . . . . . 9 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( ( A i^i ( Base ` W ) ) i^i B )
45		simpr2 1003	. . . . . . . . . . . 12 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> A e. X )
46	45, 11	syl 16	. . . . . . . . . . 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 457	. . . . . . . . . . 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 3538	. . . . . . . . . 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 3490	. . . . . . . . . 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 196	. . . . . . . . 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 2521	. . . . . . . 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 6298	. . . . . . 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 14547	. . . . . . . 8 |- ( A e. X -> ( W ↾s A ) = ( W ↾s ( A i^i ( Base ` W ) ) ) )
54	45, 53	syl 16	. . . . . . 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 14547	. . . . . . . 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, 34, 55	3syl 20	. . . . . . 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 2518	. . . . . 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 2508	. . . . 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 434	. . . 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 14539	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ W e. _V /\ A e. X ) -> ( W ↾s A ) = W )
61	60	3adant3r3 1207	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = W )
62	61	oveq1d 6297	. . . . . 6 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s B ) )
63		inss2 3719	. . . . . . . . . . 11 |- ( B i^i ( Base ` W ) ) C_ ( Base ` W )
64		simpl 457	. . . . . . . . . . 11 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( Base ` W ) C_ A )
65	63, 64	syl5ss 3515	. . . . . . . . . 10 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( Base ` W ) ) C_ A )
66		sseqin2 3717	. . . . . . . . . 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 196	. . . . . . . . 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 2521	. . . . . . . 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 6298	. . . . . . 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 1004	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> B e. Y )
71	5	ressinbas 14547	. . . . . . . 8 |- ( B e. Y -> ( W ↾s B ) = ( W ↾s ( B i^i ( Base ` W ) ) ) )
72	70, 71	syl 16	. . . . . . 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 1003	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> A e. X )
74	73, 34, 55	3syl 20	. . . . . . 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 2518	. . . . . 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 2508	. . . . 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 434	. . . 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	40, 59, 77	pm2.61ii 165	. . 3 |- ( ( W e. _V /\ A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
79	78	3expib 1199	. 2 |- ( W e. _V -> ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
80		ress0 14545	. . . 4 |- ( (/) ↾s B ) = (/)
81		reldmress 14537	. . . . . 6 |- Rel dom ↾s
82	81	ovprc1 6310	. . . . 5 |- ( -. W e. _V -> ( W ↾s A ) = (/) )
83	82	oveq1d 6297	. . . 4 |- ( -. W e. _V -> ( ( W ↾s A ) ↾s B ) = ( (/) ↾s B ) )
84	81	ovprc1 6310	. . . 4 |- ( -. W e. _V -> ( W ↾s ( A i^i B ) ) = (/) )
85	80, 83, 84	3eqtr4a 2534	. . 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 164	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 369   /\ w3a 973   = wceq 1379   e. wcel 1767   _Vcvv 3113   i^i cin 3475   C_ wss 3476   (/)c0 3785   <.cop 4033   ` cfv 5586  (class class class)co 6282   ndxcnx 14483   sSet csts 14484   Basecbs 14486   ↾_s cress 14487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-nn 10533  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493

This theorem is referenced by:  ressabs  14549  xrge00  27336  xrge0slmod  27497  esumpfinvallem  27720  lmhmlnmsplit  30637