Theorem estrres 16779

Description: Any restriction of a category (as an extensible structure which is an unordered triple of ordered pairs) is an unordered triple of ordered pairs. (Contributed by AV, 15-Mar-2020.)

Hypotheses

Ref	Expression
estrres.c	|- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
estrres.b	|- ( ph -> B e. V )
estrres.h	|- ( ph -> H e. X )
estrres.x	|- ( ph -> .x. e. Y )
estrres.a	|- ( ph -> A e. U )
estrres.g	|- ( ph -> G e. W )
estrres.u	|- ( ph -> A C_ B )

Assertion

Ref	Expression
estrres	|- ( ph -> ( ( C ↾s A ) sSet <. ( Hom ` ndx ) , G >. ) = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. } )

Proof of Theorem estrres

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 |- ( C ↾s A ) e. _V
2		estrres.g	. . 3 |- ( ph -> G e. W )
3		setsval 15888	. . 3 |- ( ( ( C ↾s A ) e. _V /\ G e. W ) -> ( ( C ↾s A ) sSet <. ( Hom ` ndx ) , G >. ) = ( ( ( C ↾s A ) |` ( _V \ { ( Hom ` ndx ) } ) ) u. { <. ( Hom ` ndx ) , G >. } ) )
4	1, 2, 3	sylancr 695	. 2 |- ( ph -> ( ( C ↾s A ) sSet <. ( Hom ` ndx ) , G >. ) = ( ( ( C ↾s A ) |` ( _V \ { ( Hom ` ndx ) } ) ) u. { <. ( Hom ` ndx ) , G >. } ) )
5		eqid 2622	. . . . . 6 |- ( C ↾s A ) = ( C ↾s A )
6		eqid 2622	. . . . . 6 |- ( Base ` C ) = ( Base ` C )
7		eqid 2622	. . . . . 6 |- ( Base ` ndx ) = ( Base ` ndx )
8		estrres.c	. . . . . . 7 |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
9		tpex 6957	. . . . . . 7 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } e. _V
10	8, 9	syl6eqel 2709	. . . . . 6 |- ( ph -> C e. _V )
11		fvex 6201	. . . . . . . . . 10 |- ( Base ` ndx ) e. _V
12		fvex 6201	. . . . . . . . . 10 |- ( Hom ` ndx ) e. _V
13		fvex 6201	. . . . . . . . . 10 |- (comp ` ndx ) e. _V
14	11, 12, 13	3pm3.2i 1239	. . . . . . . . 9 |- ( ( Base ` ndx ) e. _V /\ ( Hom ` ndx ) e. _V /\ (comp ` ndx ) e. _V )
15	14	a1i 11	. . . . . . . 8 |- ( ph -> ( ( Base ` ndx ) e. _V /\ ( Hom ` ndx ) e. _V /\ (comp ` ndx ) e. _V ) )
16		estrres.b	. . . . . . . 8 |- ( ph -> B e. V )
17		estrres.h	. . . . . . . 8 |- ( ph -> H e. X )
18		estrres.x	. . . . . . . 8 |- ( ph -> .x. e. Y )
19		slotsbhcdif 16080	. . . . . . . . 9 |- ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) )
20	19	a1i 11	. . . . . . . 8 |- ( ph -> ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) )
21		funtpg 5942	. . . . . . . 8 |- ( ( ( ( Base ` ndx ) e. _V /\ ( Hom ` ndx ) e. _V /\ (comp ` ndx ) e. _V ) /\ ( B e. V /\ H e. X /\ .x. e. Y ) /\ ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
22	15, 16, 17, 18, 20, 21	syl131anc 1339	. . . . . . 7 |- ( ph -> Fun { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
23	8	funeqd 5910	. . . . . . 7 |- ( ph -> ( Fun C <-> Fun { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } ) )
24	22, 23	mpbird 247	. . . . . 6 |- ( ph -> Fun C )
25	8, 16, 17, 18	estrreslem2 16778	. . . . . 6 |- ( ph -> ( Base ` ndx ) e. dom C )
26		estrres.a	. . . . . 6 |- ( ph -> A e. U )
27		estrres.u	. . . . . . 7 |- ( ph -> A C_ B )
28	8, 16	estrreslem1 16777	. . . . . . 7 |- ( ph -> B = ( Base ` C ) )
29	27, 28	sseqtrd 3641	. . . . . 6 |- ( ph -> A C_ ( Base ` C ) )
30	5, 6, 7, 10, 24, 25, 26, 29	ressval3d 15937	. . . . 5 |- ( ph -> ( C ↾s A ) = ( C sSet <. ( Base ` ndx ) , A >. ) )
31	30	reseq1d 5395	. . . 4 |- ( ph -> ( ( C ↾s A ) |` ( _V \ { ( Hom ` ndx ) } ) ) = ( ( C sSet <. ( Base ` ndx ) , A >. ) |` ( _V \ { ( Hom ` ndx ) } ) ) )
32	31	uneq1d 3766	. . 3 |- ( ph -> ( ( ( C ↾s A ) |` ( _V \ { ( Hom ` ndx ) } ) ) u. { <. ( Hom ` ndx ) , G >. } ) = ( ( ( C sSet <. ( Base ` ndx ) , A >. ) |` ( _V \ { ( Hom ` ndx ) } ) ) u. { <. ( Hom ` ndx ) , G >. } ) )
33		setsval 15888	. . . . . . . 8 |- ( ( C e. _V /\ A e. U ) -> ( C sSet <. ( Base ` ndx ) , A >. ) = ( ( C |` ( _V \ { ( Base ` ndx ) } ) ) u. { <. ( Base ` ndx ) , A >. } ) )
34	10, 26, 33	syl2anc 693	. . . . . . 7 |- ( ph -> ( C sSet <. ( Base ` ndx ) , A >. ) = ( ( C |` ( _V \ { ( Base ` ndx ) } ) ) u. { <. ( Base ` ndx ) , A >. } ) )
35	34	reseq1d 5395	. . . . . 6 |- ( ph -> ( ( C sSet <. ( Base ` ndx ) , A >. ) |` ( _V \ { ( Hom ` ndx ) } ) ) = ( ( ( C |` ( _V \ { ( Base ` ndx ) } ) ) u. { <. ( Base ` ndx ) , A >. } ) |` ( _V \ { ( Hom ` ndx ) } ) ) )
36	12	a1i 11	. . . . . . . . . 10 |- ( ph -> ( Hom ` ndx ) e. _V )
37	13	a1i 11	. . . . . . . . . 10 |- ( ph -> (comp ` ndx ) e. _V )
38		elex 3212	. . . . . . . . . . 11 |- ( H e. X -> H e. _V )
39	17, 38	syl 17	. . . . . . . . . 10 |- ( ph -> H e. _V )
40		elex 3212	. . . . . . . . . . 11 |- ( .x. e. Y -> .x. e. _V )
41	18, 40	syl 17	. . . . . . . . . 10 |- ( ph -> .x. e. _V )
42		simp1 1061	. . . . . . . . . . . 12 |- ( ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) -> ( Base ` ndx ) =/= ( Hom ` ndx ) )
43	42	necomd 2849	. . . . . . . . . . 11 |- ( ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) -> ( Hom ` ndx ) =/= ( Base ` ndx ) )
44	19, 43	mp1i 13	. . . . . . . . . 10 |- ( ph -> ( Hom ` ndx ) =/= ( Base ` ndx ) )
45		simp2 1062	. . . . . . . . . . . 12 |- ( ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) -> ( Base ` ndx ) =/= (comp ` ndx ) )
46	45	necomd 2849	. . . . . . . . . . 11 |- ( ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) -> (comp ` ndx ) =/= ( Base ` ndx ) )
47	19, 46	mp1i 13	. . . . . . . . . 10 |- ( ph -> (comp ` ndx ) =/= ( Base ` ndx ) )
48	8, 36, 37, 39, 41, 44, 47	tpres 6466	. . . . . . . . 9 |- ( ph -> ( C |` ( _V \ { ( Base ` ndx ) } ) ) = { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
49	48	uneq1d 3766	. . . . . . . 8 |- ( ph -> ( ( C |` ( _V \ { ( Base ` ndx ) } ) ) u. { <. ( Base ` ndx ) , A >. } ) = ( { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } u. { <. ( Base ` ndx ) , A >. } ) )
50		df-tp 4182	. . . . . . . 8 |- { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } = ( { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } u. { <. ( Base ` ndx ) , A >. } )
51	49, 50	syl6eqr 2674	. . . . . . 7 |- ( ph -> ( ( C |` ( _V \ { ( Base ` ndx ) } ) ) u. { <. ( Base ` ndx ) , A >. } ) = { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } )
52	11	a1i 11	. . . . . . 7 |- ( ph -> ( Base ` ndx ) e. _V )
53	16, 27	ssexd 4805	. . . . . . 7 |- ( ph -> A e. _V )
54		simp3 1063	. . . . . . . . 9 |- ( ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) -> ( Hom ` ndx ) =/= (comp ` ndx ) )
55	54	necomd 2849	. . . . . . . 8 |- ( ( ( Base ` ndx ) =/= ( Hom ` ndx ) /\ ( Base ` ndx ) =/= (comp ` ndx ) /\ ( Hom ` ndx ) =/= (comp ` ndx ) ) -> (comp ` ndx ) =/= ( Hom ` ndx ) )
56	19, 55	mp1i 13	. . . . . . 7 |- ( ph -> (comp ` ndx ) =/= ( Hom ` ndx ) )
57	19, 42	mp1i 13	. . . . . . 7 |- ( ph -> ( Base ` ndx ) =/= ( Hom ` ndx ) )
58	51, 37, 52, 41, 53, 56, 57	tpres 6466	. . . . . 6 |- ( ph -> ( ( ( C |` ( _V \ { ( Base ` ndx ) } ) ) u. { <. ( Base ` ndx ) , A >. } ) |` ( _V \ { ( Hom ` ndx ) } ) ) = { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } )
59	35, 58	eqtrd 2656	. . . . 5 |- ( ph -> ( ( C sSet <. ( Base ` ndx ) , A >. ) |` ( _V \ { ( Hom ` ndx ) } ) ) = { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } )
60	59	uneq1d 3766	. . . 4 |- ( ph -> ( ( ( C sSet <. ( Base ` ndx ) , A >. ) |` ( _V \ { ( Hom ` ndx ) } ) ) u. { <. ( Hom ` ndx ) , G >. } ) = ( { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } u. { <. ( Hom ` ndx ) , G >. } ) )
61		df-tp 4182	. . . . . 6 |- { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. } = ( { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } u. { <. ( Hom ` ndx ) , G >. } )
62		tprot 4284	. . . . . 6 |- { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. } = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. }
63	61, 62	eqtr3i 2646	. . . . 5 |- ( { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } u. { <. ( Hom ` ndx ) , G >. } ) = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. }
64	63	a1i 11	. . . 4 |- ( ph -> ( { <. (comp ` ndx ) , .x. >. , <. ( Base ` ndx ) , A >. } u. { <. ( Hom ` ndx ) , G >. } ) = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. } )
65	60, 64	eqtrd 2656	. . 3 |- ( ph -> ( ( ( C sSet <. ( Base ` ndx ) , A >. ) |` ( _V \ { ( Hom ` ndx ) } ) ) u. { <. ( Hom ` ndx ) , G >. } ) = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. } )
66	32, 65	eqtrd 2656	. 2 |- ( ph -> ( ( ( C ↾s A ) |` ( _V \ { ( Hom ` ndx ) } ) ) u. { <. ( Hom ` ndx ) , G >. } ) = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. } )
67	4, 66	eqtrd 2656	1 |- ( ph -> ( ( C ↾s A ) sSet <. ( Hom ` ndx ) , G >. ) = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. } )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   |` cres 5116   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858   Hom chom 15952  compcco 15953

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-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-hom 15966  df-cco 15967

This theorem is referenced by:  dfrngc2  41972  dfringc2  42018  rngcresringcat  42030