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Theorem ssc2 13715
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 2297 . . . . . 6  |-  ( ph  ->  dom  dom  J  =  dom  dom  J )
63, 5sscfn2 13711 . . . . 5  |-  ( ph  ->  J  Fn  ( dom 
dom  J  X.  dom  dom  J ) )
7 sscrel 13706 . . . . . . . 8  |-  Rel  C_cat
87brrelex2i 4746 . . . . . . 7  |-  ( H 
C_cat  J  ->  J  e.  _V )
93, 8syl 15 . . . . . 6  |-  ( ph  ->  J  e.  _V )
10 dmexg 4955 . . . . . 6  |-  ( J  e.  _V  ->  dom  J  e.  _V )
11 dmexg 4955 . . . . . 6  |-  ( dom 
J  e.  _V  ->  dom 
dom  J  e.  _V )
129, 10, 113syl 18 . . . . 5  |-  ( ph  ->  dom  dom  J  e.  _V )
134, 6, 12isssc 13713 . . . 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 5881 . . . 4  |-  ( x  =  X  ->  (
x H y )  =  ( X H y ) )
17 oveq1 5881 . . . 4  |-  ( x  =  X  ->  (
x J y )  =  ( X J y ) )
1816, 17sseq12d 3220 . . 3  |-  ( x  =  X  ->  (
( x H y )  C_  ( x J y )  <->  ( X H y )  C_  ( X J y ) ) )
19 oveq2 5882 . . . 4  |-  ( y  =  Y  ->  ( X H y )  =  ( X H Y ) )
20 oveq2 5882 . . . 4  |-  ( y  =  Y  ->  ( X J y )  =  ( X J Y ) )
2119, 20sseq12d 3220 . . 3  |-  ( y  =  Y  ->  (
( X H y )  C_  ( X J y )  <->  ( X H Y )  C_  ( X J Y ) ) )
2218, 21rspc2va 2904 . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    X. cxp 4703   dom cdm 4705    Fn wfn 5266  (class class class)co 5874    C_cat cssc 13700
This theorem is referenced by:  ssctr  13718  ssceq  13719  subcss2  13733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-ixp 6834  df-ssc 13703
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