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Theorem subccatid 14876
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 2454 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 subccat.j . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
4 subcrcl 14849 . . . 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 14872 . . 3  |-  ( ph  ->  S  C_  ( Base `  C ) )
81, 2, 5, 6, 7rescbas 14862 . 2  |-  ( ph  ->  S  =  ( Base `  D ) )
91, 2, 5, 6, 7reschom 14863 . 2  |-  ( ph  ->  J  =  ( Hom  `  D ) )
10 eqid 2454 . . 3  |-  (comp `  C )  =  (comp `  C )
111, 2, 5, 6, 7, 10rescco 14865 . 2  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
12 ovex 6226 . . . 4  |-  ( C  |`cat 
J )  e.  _V
131, 12eqeltri 2538 . . 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 14874 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (  .1.  `  x )  e.  ( x J x ) )
21 eqid 2454 . . 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 1045 . . . 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 3466 . . 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 1046 . . . 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 3466 . . 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 14873 . . . 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 1078 . . . 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 3466 . . 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 14741 . 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 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 ) ) ) )  ->  y  e.  S
)
3523, 34sseldd 3466 . . 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 14873 . . . 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 1079 . . . 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 3466 . . 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 14742 . 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 14875 . 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 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 ) ) ) )  ->  z  e.  S
)
4223, 41sseldd 3466 . . 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 14873 . . . 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 1080 . . . 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 3466 . . 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 14744 . 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 14739 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 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437    |-> cmpt 4459    X. cxp 4947    Fn wfn 5522   ` cfv 5527  (class class class)co 6201   Basecbs 14293   Hom chom 14369  compcco 14370   Catccat 14722   Idccid 14723    |`cat cresc 14841  Subcatcsubc 14842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-hom 14382  df-cco 14383  df-cat 14726  df-cid 14727  df-homf 14728  df-ssc 14843  df-resc 14844  df-subc 14845
This theorem is referenced by:  subcid  14877  subccat  14878
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