Theorem ressress 14231

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 749	. . . . . . . . 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 989	. . . . . . . . 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 990	. . . . . . . . 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 2441	. . . . . . . . . 10 |- ( W ↾s A ) = ( W ↾s A )
5		eqid 2441	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
6	4, 5	ressval2 14223	. . . . . . . . 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 1213	. . . . . . . 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 3557	. . . . . . . . . . 11 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
9		in12 3558	. . . . . . . . . . 11 |- ( A i^i ( B i^i ( Base ` W ) ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
10	8, 9	eqtri 2461	. . . . . . . . . 10 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
11	4, 5	ressbas 14224	. . . . . . . . . . . 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 3549	. . . . . . . . . 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 2486	. . . . . . . . 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 4063	. . . . . . . 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 6108	. . . . . . 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 5698	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	17	inex2 4431	. . . . . . . 8 |- ( ( A i^i B ) i^i ( Base ` W ) ) e. _V
19		setsabs 14199	. . . . . . . 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 657	. . . . . . 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 2473	. . . . . 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 748	. . . . . . 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 6115	. . . . . . . 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 991	. . . . . . 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 2441	. . . . . . . 8 |- ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) ↾s B )
27		eqid 2441	. . . . . . . 8 |- ( Base ` ( W ↾s A ) ) = ( Base ` ( W ↾s A ) )
28	26, 27	ressval2 14223	. . . . . . 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 1213	. . . . . 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 3567	. . . . . . . . 9 |- ( A i^i B ) C_ A
31		sstr 3361	. . . . . . . . 9 |- ( ( ( Base ` W ) C_ ( A i^i B ) /\ ( A i^i B ) C_ A ) -> ( Base ` W ) C_ A )
32	30, 31	mpan2 666	. . . . . . . 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 4432	. . . . . . . 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 2441	. . . . . . . 8 |- ( W ↾s ( A i^i B ) ) = ( W ↾s ( A i^i B ) )
37	36, 5	ressval2 14223	. . . . . . 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 1213	. . . . . 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 2483	. . . . 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 601	. . . 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 14222	. . . . . . . 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 1297	. . . . . . 7 |- ( ( ( Base ` ( W ↾s A ) ) C_ B /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s A ) )
43	42	3ad2antr3 1150	. . . . . 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 3559	. . . . . . . . 9 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( ( A i^i ( Base ` W ) ) i^i B )
45		simpr2 990	. . . . . . . . . . . 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 454	. . . . . . . . . . 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 3387	. . . . . . . . . 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 3339	. . . . . . . . . 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 2486	. . . . . . . 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 6106	. . . . . . 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 14230	. . . . . . . 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 14230	. . . . . . . 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 2483	. . . . . 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 2473	. . . . 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 14222	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ W e. _V /\ A e. X ) -> ( W ↾s A ) = W )
61	60	3adant3r3 1193	. . . . . . 7 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( W ↾s A ) = W )
62	61	oveq1d 6105	. . . . . 6 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s B ) )
63		inss2 3568	. . . . . . . . . . 11 |- ( B i^i ( Base ` W ) ) C_ ( Base ` W )
64		simpl 454	. . . . . . . . . . 11 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( Base ` W ) C_ A )
65	63, 64	syl5ss 3364	. . . . . . . . . 10 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> ( B i^i ( Base ` W ) ) C_ A )
66		sseqin2 3566	. . . . . . . . . 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 2486	. . . . . . . 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 6106	. . . . . . 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 991	. . . . . . . 8 |- ( ( ( Base ` W ) C_ A /\ ( W e. _V /\ A e. X /\ B e. Y ) ) -> B e. Y )
71	5	ressinbas 14230	. . . . . . . 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 990	. . . . . . . 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 2483	. . . . . 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 2473	. . . . 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 1185	. 2 |- ( W e. _V -> ( ( A e. X /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) ) )
80		ress0 14228	. . . 4 |- ( (/) ↾s B ) = (/)
81		reldmress 14220	. . . . . 6 |- Rel dom ↾s
82	81	ovprc1 6118	. . . . 5 |- ( -. W e. _V -> ( W ↾s A ) = (/) )
83	82	oveq1d 6105	. . . 4 |- ( -. W e. _V -> ( ( W ↾s A ) ↾s B ) = ( (/) ↾s B ) )
84	81	ovprc1 6118	. . . 4 |- ( -. W e. _V -> ( W ↾s ( A i^i B ) ) = (/) )
85	80, 83, 84	3eqtr4a 2499	. . 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 960   = wceq 1364   e. wcel 1761   _Vcvv 2970   i^i cin 3324   C_ wss 3325   (/)c0 3634   <.cop 3880   ` cfv 5415  (class class class)co 6090   ndxcnx 14167   sSet csts 14168   Basecbs 14170   ↾_s cress 14171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-i2m1 9346  ax-1ne0 9347  ax-rrecex 9350  ax-cnre 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-nn 10319  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177

This theorem is referenced by:  ressabs  14232  xrge00  26080  xrge0slmod  26248  esumpfinvallem  26459  lmhmlnmsplit  29365