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Theorem ssc1 15309
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 15301 . . . . . . 7  |-  Rel  C_cat
54brrelex2i 5030 . . . . . 6  |-  ( H 
C_cat  J  ->  J  e.  _V )
61, 5syl 16 . . . . 5  |-  ( ph  ->  J  e.  _V )
73ssclem 15307 . . . . 5  |-  ( ph  ->  ( J  e.  _V  <->  T  e.  _V ) )
86, 7mpbid 210 . . . 4  |-  ( ph  ->  T  e.  _V )
92, 3, 8isssc 15308 . . 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 210 . 2  |-  ( ph  ->  ( S  C_  T  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) )
1110simpld 457 1  |-  ( ph  ->  S  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    X. cxp 4986    Fn wfn 5565  (class class class)co 6270    C_cat cssc 15295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-ixp 7463  df-ssc 15298
This theorem is referenced by:  ssctr  15313  ssceq  15314  subcss1  15330  issubc3  15337  subsubc  15341
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