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Theorem subccatid 15066
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 2467 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 subccat.j . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
4 subcrcl 15039 . . . 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 15062 . . 3  |-  ( ph  ->  S  C_  ( Base `  C ) )
81, 2, 5, 6, 7rescbas 15052 . 2  |-  ( ph  ->  S  =  ( Base `  D ) )
91, 2, 5, 6, 7reschom 15053 . 2  |-  ( ph  ->  J  =  ( Hom  `  D ) )
10 eqid 2467 . . 3  |-  (comp `  C )  =  (comp `  C )
111, 2, 5, 6, 7, 10rescco 15055 . 2  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
12 ovex 6307 . . . 4  |-  ( C  |`cat 
J )  e.  _V
131, 12eqeltri 2551 . . 3  |-  D  e. 
_V
1413a1i 11 . 2  |-  ( ph  ->  D  e.  _V )
15 biid 236 . 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 465 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  e.  (Subcat `  C )
)
176adantr 465 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  Fn  ( S  X.  S
) )
18 simpr 461 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
19 subccatid.2 . . 3  |-  .1.  =  ( Id `  C )
2016, 17, 18, 19subcidcl 15064 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (  .1.  `  x )  e.  ( x J x ) )
21 eqid 2467 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
225adantr 465 . . 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 465 . . . 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 1053 . . . 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 3505 . . 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 1054 . . . 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 3505 . . 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 465 . . . . 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 465 . . . . 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 15063 . . . 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 1086 . . . 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 3505 . . 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 14931 . 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 1055 . . . 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 3505 . . 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 15063 . . . 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 1087 . . . 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 3505 . . 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 14932 . 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 15065 . 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 1056 . . . 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 3505 . . 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 15063 . . . 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 1088 . . . 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 3505 . . 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 14934 . 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 14929 1  |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D
)  =  ( x  e.  S  |->  (  .1.  `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476    |-> cmpt 4505    X. cxp 4997    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   Basecbs 14483   Hom chom 14559  compcco 14560   Catccat 14912   Idccid 14913    |`cat cresc 15031  Subcatcsubc 15032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-hom 14572  df-cco 14573  df-cat 14916  df-cid 14917  df-homf 14918  df-ssc 15033  df-resc 15034  df-subc 15035
This theorem is referenced by:  subcid  15067  subccat  15068
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