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Theorem subccatid 13720
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 2283 . . 3  |-  ( Base `  C )  =  (
Base `  C )
3 subccat.j . . . 4  |-  ( ph  ->  J  e.  (Subcat `  C ) )
4 subcrcl 13693 . . . 4  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
53, 4syl 15 . . 3  |-  ( ph  ->  C  e.  Cat )
6 subccatid.1 . . 3  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
73, 6, 2subcss1 13716 . . 3  |-  ( ph  ->  S  C_  ( Base `  C ) )
81, 2, 5, 6, 7rescbas 13706 . 2  |-  ( ph  ->  S  =  ( Base `  D ) )
91, 2, 5, 6, 7reschom 13707 . 2  |-  ( ph  ->  J  =  (  Hom  `  D ) )
10 eqid 2283 . . 3  |-  (comp `  C )  =  (comp `  C )
111, 2, 5, 6, 7, 10rescco 13709 . 2  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
12 ovex 5883 . . . 4  |-  ( C  |`cat 
J )  e.  _V
131, 12eqeltri 2353 . . 3  |-  D  e. 
_V
1413a1i 10 . 2  |-  ( ph  ->  D  e.  _V )
15 biid 227 . 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 451 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  e.  (Subcat `  C )
)
176adantr 451 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  J  Fn  ( S  X.  S
) )
18 simpr 447 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
19 subccatid.2 . . 3  |-  .1.  =  ( Id `  C )
2016, 17, 18, 19subcidcl 13718 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (  .1.  `  x )  e.  ( x J x ) )
21 eqid 2283 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
225adantr 451 . . 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 451 . . . 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 1012 . . . 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 3181 . . 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 1013 . . . 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 3181 . . 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 451 . . . . 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 451 . . . . 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 13717 . . . 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 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 ) ) ) )  ->  f  e.  ( w J x ) )
3230, 31sseldd 3181 . . 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 13585 . 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 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 ) ) ) )  ->  y  e.  S
)
3523, 34sseldd 3181 . . 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 13717 . . . 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 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 ) ) ) )  ->  g  e.  ( x J y ) )
3836, 37sseldd 3181 . . 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 13586 . 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 13719 . 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 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 ) ) ) )  ->  z  e.  S
)
4223, 41sseldd 3181 . . 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 13717 . . . 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 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 ) ) ) )  ->  h  e.  ( y J z ) )
4543, 44sseldd 3181 . . 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 13588 . 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 13583 1  |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D
)  =  ( x  e.  S  |->  (  .1.  `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    |`cat cresc 13685  Subcatcsubc 13686
This theorem is referenced by:  subcid  13721  subccat  13722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-homf 13572  df-ssc 13687  df-resc 13688  df-subc 13689
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