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Theorem subcssc 15083
Description: An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
subcixp.1  |-  ( ph  ->  J  e.  (Subcat `  C ) )
subcssc.h  |-  H  =  ( Hom f  `  C )
Assertion
Ref Expression
subcssc  |-  ( ph  ->  J  C_cat  H )

Proof of Theorem subcssc
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcixp.1 . . 3  |-  ( ph  ->  J  e.  (Subcat `  C ) )
2 subcssc.h . . . 4  |-  H  =  ( Hom f  `  C )
3 eqid 2443 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
4 eqid 2443 . . . 4  |-  (comp `  C )  =  (comp `  C )
5 subcrcl 15062 . . . . 5  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
61, 5syl 16 . . . 4  |-  ( ph  ->  C  e.  Cat )
7 eqidd 2444 . . . 4  |-  ( ph  ->  dom  dom  J  =  dom  dom  J )
82, 3, 4, 6, 7issubc 15081 . . 3  |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e. 
dom  dom  J ( ( ( Id `  C
) `  x )  e.  ( x J x )  /\  A. y  e.  dom  dom  J A. z  e.  dom  dom  J A. f  e.  (
x J y ) A. g  e.  ( y J z ) ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  e.  ( x J z ) ) ) ) )
91, 8mpbid 210 . 2  |-  ( ph  ->  ( J  C_cat  H  /\  A. x  e.  dom  dom  J ( ( ( Id
`  C ) `  x )  e.  ( x J x )  /\  A. y  e. 
dom  dom  J A. z  e.  dom  dom  J A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  e.  ( x J z ) ) ) )
109simpld 459 1  |-  ( ph  ->  J  C_cat  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   <.cop 4020   class class class wbr 4437   dom cdm 4989   ` cfv 5578  (class class class)co 6281  compcco 14586   Catccat 14938   Idccid 14939   Hom f chomf 14940    C_cat cssc 15053  Subcatcsubc 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-pm 7425  df-ixp 7472  df-ssc 15056  df-subc 15058
This theorem is referenced by:  subcfn  15084  subcss1  15085  subcss2  15086  issubc3  15092  subsubc  15096
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