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Theorem subcidcl 15700
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 2429 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 subcidcl.1 . . . . . 6  |-  .1.  =  ( Id `  C )
5 eqid 2429 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
6 subcrcl 15672 . . . . . . 7  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
72, 6syl 17 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
8 subcidcl.2 . . . . . 6  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
93, 4, 5, 7, 8issubc2 15692 . . . . 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 213 . . . 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 464 . . 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 458 . . . 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 2825 . . 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 17 . 2  |-  ( ph  ->  A. x  e.  S  (  .1.  `  x )  e.  ( x J x ) )
15 fveq2 5881 . . . 4  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
16 id 23 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1716, 16oveq12d 6323 . . . 4  |-  ( x  =  X  ->  (
x J x )  =  ( X J X ) )
1815, 17eleq12d 2511 . . 3  |-  ( x  =  X  ->  (
(  .1.  `  x
)  e.  ( x J x )  <->  (  .1.  `  X )  e.  ( X J X ) ) )
1918rspcv 3184 . 2  |-  ( X  e.  S  ->  ( A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  ->  (  .1.  `  X )  e.  ( X J X ) ) )
201, 14, 19sylc 62 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X J X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   <.cop 4008   class class class wbr 4426    X. cxp 4852    Fn wfn 5596   ` cfv 5601  (class class class)co 6305  compcco 15164   Catccat 15521   Idccid 15522   Hom f chomf 15523    C_cat cssc 15663  Subcatcsubc 15665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-pm 7483  df-ixp 7531  df-ssc 15666  df-subc 15668
This theorem is referenced by:  subccatid  15702  issubc3  15705  funcres  15752
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