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Theorem ssc2 13699
Description: Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
ssc2.1  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
ssc2.2  |-  ( ph  ->  H  C_cat  J )
ssc2.3  |-  ( ph  ->  X  e.  S )
ssc2.4  |-  ( ph  ->  Y  e.  S )
Assertion
Ref Expression
ssc2  |-  ( ph  ->  ( X H Y )  C_  ( X J Y ) )

Proof of Theorem ssc2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssc2.3 . 2  |-  ( ph  ->  X  e.  S )
2 ssc2.4 . 2  |-  ( ph  ->  Y  e.  S )
3 ssc2.2 . . . 4  |-  ( ph  ->  H  C_cat  J )
4 ssc2.1 . . . . 5  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
5 eqidd 2284 . . . . . 6  |-  ( ph  ->  dom  dom  J  =  dom  dom  J )
63, 5sscfn2 13695 . . . . 5  |-  ( ph  ->  J  Fn  ( dom 
dom  J  X.  dom  dom  J ) )
7 sscrel 13690 . . . . . . . 8  |-  Rel  C_cat
87brrelex2i 4730 . . . . . . 7  |-  ( H 
C_cat  J  ->  J  e.  _V )
93, 8syl 15 . . . . . 6  |-  ( ph  ->  J  e.  _V )
10 dmexg 4939 . . . . . 6  |-  ( J  e.  _V  ->  dom  J  e.  _V )
11 dmexg 4939 . . . . . 6  |-  ( dom 
J  e.  _V  ->  dom 
dom  J  e.  _V )
129, 10, 113syl 18 . . . . 5  |-  ( ph  ->  dom  dom  J  e.  _V )
134, 6, 12isssc 13697 . . . 4  |-  ( ph  ->  ( H  C_cat  J  <->  ( S  C_ 
dom  dom  J  /\  A. x  e.  S  A. y  e.  S  (
x H y ) 
C_  ( x J y ) ) ) )
143, 13mpbid 201 . . 3  |-  ( ph  ->  ( S  C_  dom  dom 
J  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) )
1514simprd 449 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) )
16 oveq1 5865 . . . 4  |-  ( x  =  X  ->  (
x H y )  =  ( X H y ) )
17 oveq1 5865 . . . 4  |-  ( x  =  X  ->  (
x J y )  =  ( X J y ) )
1816, 17sseq12d 3207 . . 3  |-  ( x  =  X  ->  (
( x H y )  C_  ( x J y )  <->  ( X H y )  C_  ( X J y ) ) )
19 oveq2 5866 . . . 4  |-  ( y  =  Y  ->  ( X H y )  =  ( X H Y ) )
20 oveq2 5866 . . . 4  |-  ( y  =  Y  ->  ( X J y )  =  ( X J Y ) )
2119, 20sseq12d 3207 . . 3  |-  ( y  =  Y  ->  (
( X H y )  C_  ( X J y )  <->  ( X H Y )  C_  ( X J Y ) ) )
2218, 21rspc2va 2891 . 2  |-  ( ( ( X  e.  S  /\  Y  e.  S
)  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) )  ->  ( X H Y )  C_  ( X J Y ) )
231, 2, 15, 22syl21anc 1181 1  |-  ( ph  ->  ( X H Y )  C_  ( X J Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    X. cxp 4687   dom cdm 4689    Fn wfn 5250  (class class class)co 5858    C_cat cssc 13684
This theorem is referenced by:  ssctr  13702  ssceq  13703  subcss2  13717
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-ixp 6818  df-ssc 13687
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