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Theorem subcidcl 13970
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 2389 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4 subcidcl.1 . . . . . 6  |-  .1.  =  ( Id `  C )
5 eqid 2389 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
6 subcrcl 13945 . . . . . . 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 13965 . . . . 5  |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  (  Homf 
`  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 202 . . . 4  |-  ( ph  ->  ( J  C_cat  (  Homf  `  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 450 . . 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 444 . . . 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 2726 . . 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 5670 . . . 4  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
16 id 20 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1716, 16oveq12d 6040 . . . 4  |-  ( x  =  X  ->  (
x J x )  =  ( X J X ) )
1815, 17eleq12d 2457 . . 3  |-  ( x  =  X  ->  (
(  .1.  `  x
)  e.  ( x J x )  <->  (  .1.  `  X )  e.  ( X J X ) ) )
1918rspcv 2993 . 2  |-  ( X  e.  S  ->  ( A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  ->  (  .1.  `  X )  e.  ( X J X ) ) )
201, 14, 19sylc 58 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X J X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   <.cop 3762   class class class wbr 4155    X. cxp 4818    Fn wfn 5391   ` cfv 5396  (class class class)co 6022  compcco 13470   Catccat 13818   Idccid 13819    Homf chomf 13820    C_cat cssc 13936  Subcatcsubc 13938
This theorem is referenced by:  subccatid  13972  issubc3  13975  funcres  14022
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-pm 6959  df-ixp 7002  df-ssc 13939  df-subc 13941
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