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Theorem subccatid 14006
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 2412 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 subccat.j . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
4 subcrcl 13979 . . . 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 14002 . . 3  |-  ( ph  ->  S  C_  ( Base `  C ) )
81, 2, 5, 6, 7rescbas 13992 . 2  |-  ( ph  ->  S  =  ( Base `  D ) )
91, 2, 5, 6, 7reschom 13993 . 2  |-  ( ph  ->  J  =  (  Hom  `  D ) )
10 eqid 2412 . . 3  |-  (comp `  C )  =  (comp `  C )
111, 2, 5, 6, 7, 10rescco 13995 . 2  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
12 ovex 6073 . . . 4  |-  ( C  |`cat 
J )  e.  _V
131, 12eqeltri 2482 . . 3  |-  D  e. 
_V
1413a1i 11 . 2  |-  ( ph  ->  D  e.  _V )
15 biid 228 . 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 452 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  e.  (Subcat `  C )
)
176adantr 452 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  Fn  ( S  X.  S
) )
18 simpr 448 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
19 subccatid.2 . . 3  |-  .1.  =  ( Id `  C )
2016, 17, 18, 19subcidcl 14004 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (  .1.  `  x )  e.  ( x J x ) )
21 eqid 2412 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
225adantr 452 . . 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 452 . . . 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 1014 . . . 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 3317 . . 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 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 ) ) ) )  ->  x  e.  S
)
2723, 26sseldd 3317 . . 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 452 . . . . 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 452 . . . . 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 14003 . . . 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 1047 . . . 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 3317 . . 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 13871 . 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 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 ) ) ) )  ->  y  e.  S
)
3523, 34sseldd 3317 . . 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 14003 . . . 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 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 ) ) ) )  ->  g  e.  ( x J y ) )
3836, 37sseldd 3317 . . 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 13872 . 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 14005 . 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 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 ) ) ) )  ->  z  e.  S
)
4223, 41sseldd 3317 . . 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 14003 . . . 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 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 ) ) ) )  ->  h  e.  ( y J z ) )
4543, 44sseldd 3317 . . 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 13874 . 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 13869 1  |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D
)  =  ( x  e.  S  |->  (  .1.  `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288    e. cmpt 4234    X. cxp 4843    Fn wfn 5416   ` cfv 5421  (class class class)co 6048   Basecbs 13432    Hom chom 13503  compcco 13504   Catccat 13852   Idccid 13853    |`cat cresc 13971  Subcatcsubc 13972
This theorem is referenced by:  subcid  14007  subccat  14008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-hom 13516  df-cco 13517  df-cat 13856  df-cid 13857  df-homf 13858  df-ssc 13973  df-resc 13974  df-subc 13975
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