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Theorem subcidcl 14031
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 2435 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4 subcidcl.1 . . . . . 6  |-  .1.  =  ( Id `  C )
5 eqid 2435 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
6 subcrcl 14006 . . . . . . 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 14026 . . . . 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 2773 . . 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 5720 . . . 4  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
16 id 20 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1716, 16oveq12d 6091 . . . 4  |-  ( x  =  X  ->  (
x J x )  =  ( X J X ) )
1815, 17eleq12d 2503 . . 3  |-  ( x  =  X  ->  (
(  .1.  `  x
)  e.  ( x J x )  <->  (  .1.  `  X )  e.  ( X J X ) ) )
1918rspcv 3040 . 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 1652    e. wcel 1725   A.wral 2697   <.cop 3809   class class class wbr 4204    X. cxp 4868    Fn wfn 5441   ` cfv 5446  (class class class)co 6073  compcco 13531   Catccat 13879   Idccid 13880    Homf chomf 13881    C_cat cssc 13997  Subcatcsubc 13999
This theorem is referenced by:  subccatid  14033  issubc3  14036  funcres  14083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-pm 7013  df-ixp 7056  df-ssc 14000  df-subc 14002
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