MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subcidcl Unicode version

Theorem subcidcl 13718
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 2283 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4 subcidcl.1 . . . . . 6  |-  .1.  =  ( Id `  C )
5 eqid 2283 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
6 subcrcl 13693 . . . . . . 7  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
72, 6syl 15 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
8 subcidcl.2 . . . . . 6  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
93, 4, 5, 7, 8issubc2 13713 . . . . 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 201 . . . 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 449 . . 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 443 . . . 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 2618 . . 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 15 . 2  |-  ( ph  ->  A. x  e.  S  (  .1.  `  x )  e.  ( x J x ) )
15 fveq2 5525 . . . 4  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
16 id 19 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1716, 16oveq12d 5876 . . . 4  |-  ( x  =  X  ->  (
x J x )  =  ( X J X ) )
1815, 17eleq12d 2351 . . 3  |-  ( x  =  X  ->  (
(  .1.  `  x
)  e.  ( x J x )  <->  (  .1.  `  X )  e.  ( X J X ) ) )
1918rspcv 2880 . 2  |-  ( X  e.  S  ->  ( A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  ->  (  .1.  `  X )  e.  ( X J X ) ) )
201, 14, 19sylc 56 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X J X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858  compcco 13220   Catccat 13566   Idccid 13567    Homf chomf 13568    C_cat cssc 13684  Subcatcsubc 13686
This theorem is referenced by:  subccatid  13720  issubc3  13723  funcres  13770
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-pm 6775  df-ixp 6818  df-ssc 13687  df-subc 13689
  Copyright terms: Public domain W3C validator