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Theorem resscatc 16755
Description: The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat ` U categories for different U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resscatc.c |- C = (CatCat ` U )
resscatc.d |- D = (CatCat ` V )
resscatc.1 |- ( ph -> U e. W )
resscatc.2 |- ( ph -> V C_ U )
Assertion
Ref Expression
resscatc |- ( ph -> ( ( Hom f ` ( C ↾s V ) ) = ( Hom f ` D ) /\ (compf ` ( C ↾s V ) ) = (compf ` D ) ) )
Proof of Theorem resscatc
Dummy variables f g x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resscatc.d . . . . . 6 |- D = (CatCat ` V )
2 eqid 2622 . . . . . 6 |- ( Base ` D ) = ( Base ` D )
3 resscatc.1 . . . . . . . 8 |- ( ph -> U e. W )
4 resscatc.2 . . . . . . . 8 |- ( ph -> V C_ U )
53, 4ssexd 4805 . . . . . . 7 |- ( ph -> V e. _V )
65adantr 481 . . . . . 6 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> V e. _V )
7 eqid 2622 . . . . . 6 |- ( Hom ` D ) = ( Hom ` D )
8 simprl 794 . . . . . . 7 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> x e. ( V i^i Cat ) )
91, 2, 5catcbas 16747 . . . . . . . 8 |- ( ph -> ( Base ` D ) = ( V i^i Cat ) )
109adantr 481 . . . . . . 7 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( Base ` D ) = ( V i^i Cat ) )
118, 10eleqtrrd 2704 . . . . . 6 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> x e. ( Base ` D ) )
12 simprr 796 . . . . . . 7 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> y e. ( V i^i Cat ) )
1312, 10eleqtrrd 2704 . . . . . 6 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> y e. ( Base ` D ) )
141, 2, 6, 7, 11, 13catchom 16749 . . . . 5 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` D ) y ) = ( x Func y ) )
15 resscatc.c . . . . . 6 |- C = (CatCat ` U )
16 eqid 2622 . . . . . 6 |- ( Base ` C ) = ( Base ` C )
173adantr 481 . . . . . 6 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> U e. W )
18 eqid 2622 . . . . . 6 |- ( Hom ` C ) = ( Hom ` C )
1915, 16, 3catcbas 16747 . . . . . . . . . . . 12 |- ( ph -> ( Base ` C ) = ( U i^i Cat ) )
2019ineq2d 3814 . . . . . . . . . . 11 |- ( ph -> ( V i^i ( Base ` C ) ) = ( V i^i ( U i^i Cat ) ) )
21 inass 3823 . . . . . . . . . . 11 |- ( ( V i^i U ) i^i Cat ) = ( V i^i ( U i^i Cat ) )
2220, 21syl6reqr 2675 . . . . . . . . . 10 |- ( ph -> ( ( V i^i U ) i^i Cat ) = ( V i^i ( Base ` C ) ) )
23 df-ss 3588 . . . . . . . . . . . 12 |- ( V C_ U <-> ( V i^i U ) = V )
244, 23sylib 208 . . . . . . . . . . 11 |- ( ph -> ( V i^i U ) = V )
2524ineq1d 3813 . . . . . . . . . 10 |- ( ph -> ( ( V i^i U ) i^i Cat ) = ( V i^i Cat ) )
26 eqid 2622 . . . . . . . . . . . 12 |- ( C ↾s V ) = ( C ↾s V )
2726, 16ressbas 15930 . . . . . . . . . . 11 |- ( V e. _V -> ( V i^i ( Base ` C ) ) = ( Base ` ( C ↾s V ) ) )
285, 27syl 17 . . . . . . . . . 10 |- ( ph -> ( V i^i ( Base ` C ) ) = ( Base ` ( C ↾s V ) ) )
2922, 25, 283eqtr3d 2664 . . . . . . . . 9 |- ( ph -> ( V i^i Cat ) = ( Base ` ( C ↾s V ) ) )
3026, 16ressbasss 15932 . . . . . . . . 9 |- ( Base ` ( C ↾s V ) ) C_ ( Base ` C )
3129, 30syl6eqss 3655 . . . . . . . 8 |- ( ph -> ( V i^i Cat ) C_ ( Base ` C ) )
3231adantr 481 . . . . . . 7 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( V i^i Cat ) C_ ( Base ` C ) )
3332, 8sseldd 3604 . . . . . 6 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> x e. ( Base ` C ) )
3432, 12sseldd 3604 . . . . . 6 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> y e. ( Base ` C ) )
3515, 16, 17, 18, 33, 34catchom 16749 . . . . 5 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` C ) y ) = ( x Func y ) )
3626, 18resshom 16078 . . . . . . 7 |- ( V e. _V -> ( Hom ` C ) = ( Hom ` ( C ↾s V ) ) )
375, 36syl 17 . . . . . 6 |- ( ph -> ( Hom ` C ) = ( Hom ` ( C ↾s V ) ) )
3837oveqdr 6674 . . . . 5 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` C ) y ) = ( x ( Hom ` ( C ↾s V ) ) y ) )
3914, 35, 383eqtr2rd 2663 . . . 4 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` ( C ↾s V ) ) y ) = ( x ( Hom ` D ) y ) )
4039ralrimivva 2971 . . 3 |- ( ph -> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) ( x ( Hom ` ( C ↾s V ) ) y ) = ( x ( Hom ` D ) y ) )
41 eqid 2622 . . . 4 |- ( Hom ` ( C ↾s V ) ) = ( Hom ` ( C ↾s V ) )
429eqcomd 2628 . . . 4 |- ( ph -> ( V i^i Cat ) = ( Base ` D ) )
4341, 7, 29, 42homfeq 16354 . . 3 |- ( ph -> ( ( Hom f ` ( C ↾s V ) ) = ( Hom f ` D ) <-> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) ( x ( Hom ` ( C ↾s V ) ) y ) = ( x ( Hom ` D ) y ) ) )
4440, 43mpbird 247 . 2 |- ( ph -> ( Hom f ` ( C ↾s V ) ) = ( Hom f ` D ) )
455ad2antrr 762 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> V e. _V )
46 eqid 2622 . . . . . . . 8 |- (comp ` D ) = (comp ` D )
47 simplr1 1103 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> x e. ( V i^i Cat ) )
489ad2antrr 762 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( Base ` D ) = ( V i^i Cat ) )
4947, 48eleqtrrd 2704 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> x e. ( Base ` D ) )
50 simplr2 1104 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> y e. ( V i^i Cat ) )
5150, 48eleqtrrd 2704 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> y e. ( Base ` D ) )
52 simplr3 1105 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> z e. ( V i^i Cat ) )
5352, 48eleqtrrd 2704 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> z e. ( Base ` D ) )
54 simprl 794 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> f e. ( x ( Hom ` D ) y ) )
551, 2, 45, 7, 49, 51catchom 16749 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( x ( Hom ` D ) y ) = ( x Func y ) )
5654, 55eleqtrd 2703 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> f e. ( x Func y ) )
57 simprr 796 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> g e. ( y ( Hom ` D ) z ) )
581, 2, 45, 7, 51, 53catchom 16749 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( y ( Hom ` D ) z ) = ( y Func z ) )
5957, 58eleqtrd 2703 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> g e. ( y Func z ) )
601, 2, 45, 46, 49, 51, 53, 56, 59catcco 16751 . . . . . . 7 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. (comp ` D ) z ) f ) = ( g o.func f ) )
613ad2antrr 762 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> U e. W )
62 eqid 2622 . . . . . . . 8 |- (comp ` C ) = (comp ` C )
6331ad2antrr 762 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( V i^i Cat ) C_ ( Base ` C ) )
6463, 47sseldd 3604 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> x e. ( Base ` C ) )
6563, 50sseldd 3604 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> y e. ( Base ` C ) )
6663, 52sseldd 3604 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> z e. ( Base ` C ) )
6715, 16, 61, 62, 64, 65, 66, 56, 59catcco 16751 . . . . . . 7 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. (comp ` C ) z ) f ) = ( g o.func f ) )
6826, 62ressco 16079 . . . . . . . . . . 11 |- ( V e. _V -> (comp ` C ) = (comp ` ( C ↾s V ) ) )
695, 68syl 17 . . . . . . . . . 10 |- ( ph -> (comp ` C ) = (comp ` ( C ↾s V ) ) )
7069ad2antrr 762 . . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> (comp ` C ) = (comp ` ( C ↾s V ) ) )
7170oveqd 6667 . . . . . . . 8 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( <. x , y >. (comp ` C ) z ) = ( <. x , y >. (comp ` ( C ↾s V ) ) z ) )
7271oveqd 6667 . . . . . . 7 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. (comp ` C ) z ) f ) = ( g ( <. x , y >. (comp ` ( C ↾s V ) ) z ) f ) )
7360, 67, 723eqtr2d 2662 . . . . . 6 |- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. (comp ` D ) z ) f ) = ( g ( <. x , y >. (comp ` ( C ↾s V ) ) z ) f ) )
7473ralrimivva 2971 . . . . 5 |- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) -> A. f e. ( x ( Hom ` D ) y ) A. g e. ( y ( Hom ` D ) z ) ( g ( <. x , y >. (comp ` D ) z ) f ) = ( g ( <. x , y >. (comp ` ( C ↾s V ) ) z ) f ) )
7574ralrimivvva 2972 . . . 4 |- ( ph -> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) A. z e. ( V i^i Cat ) A. f e. ( x ( Hom ` D ) y ) A. g e. ( y ( Hom ` D ) z ) ( g ( <. x , y >. (comp ` D ) z ) f ) = ( g ( <. x , y >. (comp ` ( C ↾s V ) ) z ) f ) )
76 eqid 2622 . . . . 5 |- (comp ` ( C ↾s V ) ) = (comp ` ( C ↾s V ) )
7744eqcomd 2628 . . . . 5 |- ( ph -> ( Hom f ` D ) = ( Hom f ` ( C ↾s V ) ) )
7846, 76, 7, 42, 29, 77comfeq 16366 . . . 4 |- ( ph -> ( (compf ` D ) = (compf ` ( C ↾s V ) ) <-> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) A. z e. ( V i^i Cat ) A. f e. ( x ( Hom ` D ) y ) A. g e. ( y ( Hom ` D ) z ) ( g ( <. x , y >. (comp ` D ) z ) f ) = ( g ( <. x , y >. (comp ` ( C ↾s V ) ) z ) f ) ) )
7975, 78mpbird 247 . . 3 |- ( ph -> (compf ` D ) = (compf ` ( C ↾s V ) ) )
8079eqcomd 2628 . 2 |- ( ph -> (compf ` ( C ↾s V ) ) = (compf ` D ) )
8144, 80jca 554 1 |- ( ph -> ( ( Hom f ` ( C ↾s V ) ) = ( Hom f ` D ) /\ (compf ` ( C ↾s V ) ) = (compf ` D ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾s cress 15858   Hom chom 15952  compcco 15953   Catccat 16325   Hom f chomf 16327  compfccomf 16328   Func cfunc 16514   o.func ccofu 16516  CatCatccatc 16744
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-rep 4771  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-int 4476  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-hom 15966  df-cco 15967  df-homf 16331  df-comf 16332  df-catc 16745
This theorem is referenced by: (None)
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