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Theorem subccatid 14074
Description: A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subccat.1  |-  D  =  ( C  |`cat  J )
subccat.j  |-  ( ph  ->  J  e.  (Subcat `  C ) )
subccatid.1  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
subccatid.2  |-  .1.  =  ( Id `  C )
Assertion
Ref Expression
subccatid  |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D
)  =  ( x  e.  S  |->  (  .1.  `  x ) ) ) )
Distinct variable groups:    x, C    x, D    ph, x    x,  .1.    x, J    x, S

Proof of Theorem subccatid
Dummy variables  f 
g  h  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subccat.1 . . 3  |-  D  =  ( C  |`cat  J )
2 eqid 2442 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 subccat.j . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
4 subcrcl 14047 . . . 4  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
53, 4syl 16 . . 3  |-  ( ph  ->  C  e.  Cat )
6 subccatid.1 . . 3  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
73, 6, 2subcss1 14070 . . 3  |-  ( ph  ->  S  C_  ( Base `  C ) )
81, 2, 5, 6, 7rescbas 14060 . 2  |-  ( ph  ->  S  =  ( Base `  D ) )
91, 2, 5, 6, 7reschom 14061 . 2  |-  ( ph  ->  J  =  (  Hom  `  D ) )
10 eqid 2442 . . 3  |-  (comp `  C )  =  (comp `  C )
111, 2, 5, 6, 7, 10rescco 14063 . 2  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
12 ovex 6135 . . . 4  |-  ( C  |`cat 
J )  e.  _V
131, 12eqeltri 2512 . . 3  |-  D  e. 
_V
1413a1i 11 . 2  |-  ( ph  ->  D  e.  _V )
15 biid 229 . 2  |-  ( ( ( w  e.  S  /\  x  e.  S
)  /\  ( y  e.  S  /\  z  e.  S )  /\  (
f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) )  <->  ( ( w  e.  S  /\  x  e.  S )  /\  (
y  e.  S  /\  z  e.  S )  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )
163adantr 453 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  e.  (Subcat `  C )
)
176adantr 453 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  Fn  ( S  X.  S
) )
18 simpr 449 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
19 subccatid.2 . . 3  |-  .1.  =  ( Id `  C )
2016, 17, 18, 19subcidcl 14072 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (  .1.  `  x )  e.  ( x J x ) )
21 eqid 2442 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
225adantr 453 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  C  e.  Cat )
237adantr 453 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  S  C_  ( Base `  C ) )
24 simpr1l 1015 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  w  e.  S
)
2523, 24sseldd 3335 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  w  e.  (
Base `  C )
)
26 simpr1r 1016 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  x  e.  S
)
2723, 26sseldd 3335 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  x  e.  (
Base `  C )
)
283adantr 453 . . . . 5  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  J  e.  (Subcat `  C ) )
296adantr 453 . . . . 5  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  J  Fn  ( S  X.  S ) )
3028, 29, 21, 24, 26subcss2 14071 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  ( w J x )  C_  (
w (  Hom  `  C
) x ) )
31 simpr31 1048 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  f  e.  ( w J x ) )
3230, 31sseldd 3335 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  f  e.  ( w (  Hom  `  C
) x ) )
332, 21, 19, 22, 25, 10, 27, 32catlid 13939 . 2  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  ( (  .1.  `  x ) ( <.
w ,  x >. (comp `  C ) x ) f )  =  f )
34 simpr2l 1017 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  y  e.  S
)
3523, 34sseldd 3335 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  y  e.  (
Base `  C )
)
3628, 29, 21, 26, 34subcss2 14071 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  ( x J y )  C_  (
x (  Hom  `  C
) y ) )
37 simpr32 1049 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  g  e.  ( x J y ) )
3836, 37sseldd 3335 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  g  e.  ( x (  Hom  `  C
) y ) )
392, 21, 19, 22, 27, 10, 35, 38catrid 13940 . 2  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  ( g (
<. x ,  x >. (comp `  C ) y ) (  .1.  `  x
) )  =  g )
4028, 29, 24, 10, 26, 34, 31, 37subccocl 14073 . 2  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  ( g (
<. w ,  x >. (comp `  C ) y ) f )  e.  ( w J y ) )
41 simpr2r 1018 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  z  e.  S
)
4223, 41sseldd 3335 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  z  e.  (
Base `  C )
)
4328, 29, 21, 34, 41subcss2 14071 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  ( y J z )  C_  (
y (  Hom  `  C
) z ) )
44 simpr33 1050 . . . 4  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  h  e.  ( y J z ) )
4543, 44sseldd 3335 . . 3  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  h  e.  ( y (  Hom  `  C
) z ) )
462, 21, 10, 22, 25, 27, 35, 32, 38, 42, 45catass 13942 . 2  |-  ( (
ph  /\  ( (
w  e.  S  /\  x  e.  S )  /\  ( y  e.  S  /\  z  e.  S
)  /\  ( f  e.  ( w J x )  /\  g  e.  ( x J y )  /\  h  e.  ( y J z ) ) ) )  ->  ( ( h ( <. x ,  y
>. (comp `  C )
z ) g ) ( <. w ,  x >. (comp `  C )
z ) f )  =  ( h (
<. w ,  y >.
(comp `  C )
z ) ( g ( <. w ,  x >. (comp `  C )
y ) f ) ) )
478, 9, 11, 14, 15, 20, 33, 39, 40, 46iscatd2 13937 1  |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D
)  =  ( x  e.  S  |->  (  .1.  `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   _Vcvv 2962    C_ wss 3306    e. cmpt 4291    X. cxp 4905    Fn wfn 5478   ` cfv 5483  (class class class)co 6110   Basecbs 13500    Hom chom 13571  compcco 13572   Catccat 13920   Idccid 13921    |`cat cresc 14039  Subcatcsubc 14040
This theorem is referenced by:  subcid  14075  subccat  14076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-hom 13584  df-cco 13585  df-cat 13924  df-cid 13925  df-homf 13926  df-ssc 14041  df-resc 14042  df-subc 14043
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