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Theorem subcssc 14039
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  =  (  Homf 
`  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  =  (  Homf 
`  C )
3 eqid 2438 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
4 eqid 2438 . . . 4  |-  (comp `  C )  =  (comp `  C )
5 subcrcl 14018 . . . . 5  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
61, 5syl 16 . . . 4  |-  ( ph  ->  C  e.  Cat )
7 eqidd 2439 . . . 4  |-  ( ph  ->  dom  dom  J  =  dom  dom  J )
82, 3, 4, 6, 7issubc 14037 . . 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 203 . 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 447 1  |-  ( ph  ->  J  C_cat  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   <.cop 3819   class class class wbr 4214   dom cdm 4880   ` cfv 5456  (class class class)co 6083  compcco 13543   Catccat 13891   Idccid 13892    Homf chomf 13893    C_cat cssc 14009  Subcatcsubc 14011
This theorem is referenced by:  subcfn  14040  subcss1  14041  subcss2  14042  issubc3  14048  subsubc  14052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-pm 7023  df-ixp 7066  df-ssc 14012  df-subc 14014
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