Theorem ressress 14796

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 1001	. . . . . . . . 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 1002	. . . . . . . . 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 2400	. . . . . . . . . 10 |- ( W ↾s A ) = ( W ↾s A )
5		eqid 2400	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
6	4, 5	ressval2 14787	. . . . . . . . 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 3646	. . . . . . . . . . 11 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
9		in12 3647	. . . . . . . . . . 11 |- ( A i^i ( B i^i ( Base ` W ) ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
10	8, 9	eqtri 2429	. . . . . . . . . 10 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
11	4, 5	ressbas 14788	. . . . . . . . . . . 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 3638	. . . . . . . . . 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 2454	. . . . . . . . 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 4163	. . . . . . . 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 6250	. . . . . . 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 5813	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	17	inex2 4533	. . . . . . . 8 |- ( ( A i^i B ) i^i ( Base ` W ) ) e. _V
19		setsabs 14762	. . . . . . . 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 660	. . . . . . 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 2441	. . . . . 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 752	. . . . . . 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 6260	. . . . . . . 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 1003	. . . . . . 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 2400	. . . . . . . 8 |- ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) ↾s B )
27		eqid 2400	. . . . . . . 8 |- ( Base ` ( W ↾s A ) ) = ( Base ` ( W ↾s A ) )
28	26, 27	ressval2 14787	. . . . . . 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 3656	. . . . . . . . 9 |- ( A i^i B ) C_ A
31		sstr 3447	. . . . . . . . 9 |- ( ( ( Base ` W ) C_ ( A i^i B ) /\ ( A i^i B ) C_ A ) -> ( Base ` W ) C_ A )
32	30, 31	mpan2 669	. . . . . . . 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 4534	. . . . . . . 8 |- ( A e. X -> ( A i^i B ) e. _V )
35	3, 34	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 )
36		eqid 2400	. . . . . . . 8 |- ( W ↾s ( A i^i B ) ) = ( W ↾s ( A i^i B ) )
37	36, 5	ressval2 14787	. . . . . . 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 2451	. . . . 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 602	. . . 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 14786	. . . . . . . 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 1162	. . . . . 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 3648	. . . . . . . . 9 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( ( A i^i ( Base ` W ) ) i^i B )
45		simpr2 1002	. . . . . . . . . . . 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 455	. . . . . . . . . . 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 3473	. . . . . . . . . 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 3425	. . . . . . . . . 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 2454	. . . . . . . 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 6248	. . . . . . 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 14794	. . . . . . . 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 14794	. . . . . . . 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 2451	. . . . . 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 2441	. . . . 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 432	. . . 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 14786	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ W e. _V /\ A e. X ) -> ( W ↾s A ) = W )
61	60	3adant3r3 1206	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = W )
62	61	oveq1d 6247	. . . . . 6 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s B ) )
63		inss2 3657	. . . . . . . . . . 11 |- ( B i^i ( Base ` W ) ) C_ ( Base ` W )
64		simpl 455	. . . . . . . . . . 11 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( Base ` W ) C_ A )
65	63, 64	syl5ss 3450	. . . . . . . . . 10 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( Base ` W ) ) C_ A )
66		sseqin2 3655	. . . . . . . . . 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 2454	. . . . . . . 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 6248	. . . . . . 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 1003	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> B e. Y )
71	5	ressinbas 14794	. . . . . . . 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 1002	. . . . . . . 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 2451	. . . . . 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 2441	. . . . 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 432	. . . 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 1198	. 2 |- ( W e. _V -> ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
80		ress0 14792	. . . 4 |- ( (/) ↾s B ) = (/)
81		reldmress 14784	. . . . . 6 |- Rel dom ↾s
82	81	ovprc1 6263	. . . . 5 |- ( -. W e. _V -> ( W ↾s A ) = (/) )
83	82	oveq1d 6247	. . . 4 |- ( -. W e. _V -> ( ( W ↾s A ) ↾s B ) = ( (/) ↾s B ) )
84	81	ovprc1 6263	. . . 4 |- ( -. W e. _V -> ( W ↾s ( A i^i B ) ) = (/) )
85	80, 83, 84	3eqtr4a 2467	. . 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 367   /\ w3a 972   = wceq 1403   e. wcel 1840   _Vcvv 3056   i^i cin 3410   C_ wss 3411   (/)c0 3735   <.cop 3975   ` cfv 5523  (class class class)co 6232   ndxcnx 14728   sSet csts 14729   Basecbs 14731   ↾_s cress 14732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-i2m1 9508  ax-1ne0 9509  ax-rrecex 9512  ax-cnre 9513

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-recs 6997  df-rdg 7031  df-nn 10495  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738

This theorem is referenced by:  ressabs  14797  xrge00  28007  xrge0slmod  28168  esumpfinvallem  28402  lmhmlnmsplit  35359