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Theorem subcidcl 15083
Description: The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcidcl.j  |-  ( ph  ->  J  e.  (Subcat `  C ) )
subcidcl.2  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
subcidcl.x  |-  ( ph  ->  X  e.  S )
subcidcl.1  |-  .1.  =  ( Id `  C )
Assertion
Ref Expression
subcidcl  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X J X ) )

Proof of Theorem subcidcl
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcidcl.x . 2  |-  ( ph  ->  X  e.  S )
2 subcidcl.j . . . . 5  |-  ( ph  ->  J  e.  (Subcat `  C ) )
3 eqid 2467 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 subcidcl.1 . . . . . 6  |-  .1.  =  ( Id `  C )
5 eqid 2467 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
6 subcrcl 15058 . . . . . . 7  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
72, 6syl 16 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
8 subcidcl.2 . . . . . 6  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
93, 4, 5, 7, 8issubc2 15078 . . . . 5  |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  S  (
(  .1.  `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  e.  ( x J z ) ) ) ) )
102, 9mpbid 210 . . . 4  |-  ( ph  ->  ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  S  (
(  .1.  `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  e.  ( x J z ) ) ) )
1110simprd 463 . . 3  |-  ( ph  ->  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.
(comp `  C )
z ) f )  e.  ( x J z ) ) )
12 simpl 457 . . . 4  |-  ( ( (  .1.  `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  e.  ( x J z ) )  ->  (  .1.  `  x )  e.  ( x J x ) )
1312ralimi 2860 . . 3  |-  ( A. x  e.  S  (
(  .1.  `  x
)  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  e.  ( x J z ) )  ->  A. x  e.  S  (  .1.  `  x )  e.  ( x J x ) )
1411, 13syl 16 . 2  |-  ( ph  ->  A. x  e.  S  (  .1.  `  x )  e.  ( x J x ) )
15 fveq2 5871 . . . 4  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
16 id 22 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1716, 16oveq12d 6312 . . . 4  |-  ( x  =  X  ->  (
x J x )  =  ( X J X ) )
1815, 17eleq12d 2549 . . 3  |-  ( x  =  X  ->  (
(  .1.  `  x
)  e.  ( x J x )  <->  (  .1.  `  X )  e.  ( X J X ) ) )
1918rspcv 3215 . 2  |-  ( X  e.  S  ->  ( A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  ->  (  .1.  `  X )  e.  ( X J X ) ) )
201, 14, 19sylc 60 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X J X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   <.cop 4038   class class class wbr 4452    X. cxp 5002    Fn wfn 5588   ` cfv 5593  (class class class)co 6294  compcco 14579   Catccat 14931   Idccid 14932   Hom f chomf 14933    C_cat cssc 15049  Subcatcsubc 15051
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-pm 7433  df-ixp 7480  df-ssc 15052  df-subc 15054
This theorem is referenced by:  subccatid  15085  issubc3  15088  funcres  15135
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