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Theorem ssc1 13714
Description: Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
isssc.1  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
isssc.2  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
ssc1.3  |-  ( ph  ->  H  C_cat  J )
Assertion
Ref Expression
ssc1  |-  ( ph  ->  S  C_  T )

Proof of Theorem ssc1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssc1.3 . . 3  |-  ( ph  ->  H  C_cat  J )
2 isssc.1 . . . 4  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
3 isssc.2 . . . 4  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
4 sscrel 13706 . . . . . . 7  |-  Rel  C_cat
54brrelex2i 4746 . . . . . 6  |-  ( H 
C_cat  J  ->  J  e.  _V )
61, 5syl 15 . . . . 5  |-  ( ph  ->  J  e.  _V )
73ssclem 13712 . . . . 5  |-  ( ph  ->  ( J  e.  _V  <->  T  e.  _V ) )
86, 7mpbid 201 . . . 4  |-  ( ph  ->  T  e.  _V )
92, 3, 8isssc 13713 . . 3  |-  ( ph  ->  ( H  C_cat  J  <->  ( S  C_  T  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) ) )
101, 9mpbid 201 . 2  |-  ( ph  ->  ( S  C_  T  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) )
1110simpld 445 1  |-  ( ph  ->  S  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    X. cxp 4703    Fn wfn 5266  (class class class)co 5874    C_cat cssc 13700
This theorem is referenced by:  ssctr  13718  ssceq  13719  subcss1  13732  issubc3  13739  subsubc  13743
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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