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Theorem subcssc 13763
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 2316 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
4 eqid 2316 . . . 4  |-  (comp `  C )  =  (comp `  C )
5 subcrcl 13742 . . . . 5  |-  ( J  e.  (Subcat `  C
)  ->  C  e.  Cat )
61, 5syl 15 . . . 4  |-  ( ph  ->  C  e.  Cat )
7 eqidd 2317 . . . 4  |-  ( ph  ->  dom  dom  J  =  dom  dom  J )
82, 3, 4, 6, 7issubc 13761 . . 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 201 . 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 445 1  |-  ( ph  ->  J  C_cat  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   <.cop 3677   class class class wbr 4060   dom cdm 4726   ` cfv 5292  (class class class)co 5900  compcco 13267   Catccat 13615   Idccid 13616    Homf chomf 13617    C_cat cssc 13733  Subcatcsubc 13735
This theorem is referenced by:  subcfn  13764  subcss1  13765  subcss2  13766  issubc3  13772  subsubc  13776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-pm 6818  df-ixp 6861  df-ssc 13736  df-subc 13738
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