Theorem ressress 15265

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 770	. . . . . . . . 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 1036	. . . . . . . . 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 1037	. . . . . . . . 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 2471	. . . . . . . . . 10 |- ( W ↾s A ) = ( W ↾s A )
5		eqid 2471	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
6	4, 5	ressval2 15256	. . . . . . . . 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 1292	. . . . . . . 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 3633	. . . . . . . . . . 11 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
9		in12 3634	. . . . . . . . . . 11 |- ( A i^i ( B i^i ( Base ` W ) ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
10	8, 9	eqtri 2493	. . . . . . . . . 10 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
11	4, 5	ressbas 15257	. . . . . . . . . . . 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 3625	. . . . . . . . . 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 2518	. . . . . . . . 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 4165	. . . . . . . 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 6326	. . . . . . 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 5889	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	17	inex2 4538	. . . . . . . 8 |- ( ( A i^i B ) i^i ( Base ` W ) ) e. _V
19		setsabs 15230	. . . . . . . 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 675	. . . . . . 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 2505	. . . . . 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 768	. . . . . . 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 6336	. . . . . . . 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 1038	. . . . . . 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 2471	. . . . . . . 8 |- ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) ↾s B )
27		eqid 2471	. . . . . . . 8 |- ( Base ` ( W ↾s A ) ) = ( Base ` ( W ↾s A ) )
28	26, 27	ressval2 15256	. . . . . . 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 1292	. . . . . 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 3643	. . . . . . . . 9 |- ( A i^i B ) C_ A
31		sstr 3426	. . . . . . . . 9 |- ( ( ( Base ` W ) C_ ( A i^i B ) /\ ( A i^i B ) C_ A ) -> ( Base ` W ) C_ A )
32	30, 31	mpan2 685	. . . . . . . 8 |- ( ( Base ` W ) C_ ( A i^i B ) -> ( Base ` W ) C_ A )
33	1, 32	nsyl 125	. . . . . . 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 4539	. . . . . . . 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 2471	. . . . . . . 8 |- ( W ↾s ( A i^i B ) ) = ( W ↾s ( A i^i B ) )
37	36, 5	ressval2 15256	. . . . . . 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 1292	. . . . . 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 2515	. . . . 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 615	. . . 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 15255	. . . . . . . 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 1378	. . . . . . 7 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s A ) )
43	42	3ad2antr3 1197	. . . . . 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 3635	. . . . . . . . 9 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( ( A i^i ( Base ` W ) ) i^i B )
45		simpr2 1037	. . . . . . . . . . . 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 464	. . . . . . . . . . 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 3452	. . . . . . . . . 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 3404	. . . . . . . . . 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 201	. . . . . . . . 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 2518	. . . . . . . 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 6324	. . . . . . 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 15263	. . . . . . . 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 15263	. . . . . . . 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 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 2515	. . . . . 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 2505	. . . . 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 441	. . . 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 15255	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ W e. _V /\ A e. X ) -> ( W ↾s A ) = W )
61	60	3adant3r3 1242	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = W )
62	61	oveq1d 6323	. . . . . 6 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s B ) )
63		inss2 3644	. . . . . . . . . . 11 |- ( B i^i ( Base ` W ) ) C_ ( Base ` W )
64		simpl 464	. . . . . . . . . . 11 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( Base ` W ) C_ A )
65	63, 64	syl5ss 3429	. . . . . . . . . 10 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( Base ` W ) ) C_ A )
66		sseqin2 3642	. . . . . . . . . 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 201	. . . . . . . . 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 2518	. . . . . . . 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 6324	. . . . . . 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 1038	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> B e. Y )
71	5	ressinbas 15263	. . . . . . . 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 1037	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> A e. X )
74	73, 34, 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 2515	. . . . . 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 2505	. . . . 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 441	. . . 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 170	. . 3 |- ( ( W e. _V /\ A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
79	78	3expib 1234	. 2 |- ( W e. _V -> ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
80		ress0 15261	. . . 4 |- ( (/) ↾s B ) = (/)
81		reldmress 15253	. . . . . 6 |- Rel dom ↾s
82	81	ovprc1 6339	. . . . 5 |- ( -. W e. _V -> ( W ↾s A ) = (/) )
83	82	oveq1d 6323	. . . 4 |- ( -. W e. _V -> ( ( W ↾s A ) ↾s B ) = ( (/) ↾s B ) )
84	81	ovprc1 6339	. . . 4 |- ( -. W e. _V -> ( W ↾s ( A i^i B ) ) = (/) )
85	80, 83, 84	3eqtr4a 2531	. . 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 169	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 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   _Vcvv 3031   i^i cin 3389   C_ wss 3390   (/)c0 3722   <.cop 3965   ` cfv 5589  (class class class)co 6308   ndxcnx 15196   sSet csts 15197   Basecbs 15199   ↾_s cress 15200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rrecex 9629  ax-cnre 9630

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-nn 10632  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206

This theorem is referenced by:  ressabs  15266  xrge00  28523  xrge0slmod  28681  esumpfinvallem  28969  lmhmlnmsplit  36016