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Theorem ssc1 14011
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 14003 . . . . . . 7  |-  Rel  C_cat
54brrelex2i 4911 . . . . . 6  |-  ( H 
C_cat  J  ->  J  e.  _V )
61, 5syl 16 . . . . 5  |-  ( ph  ->  J  e.  _V )
73ssclem 14009 . . . . 5  |-  ( ph  ->  ( J  e.  _V  <->  T  e.  _V ) )
86, 7mpbid 202 . . . 4  |-  ( ph  ->  T  e.  _V )
92, 3, 8isssc 14010 . . 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 202 . 2  |-  ( ph  ->  ( S  C_  T  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) )
1110simpld 446 1  |-  ( ph  ->  S  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   class class class wbr 4204    X. cxp 4868    Fn wfn 5441  (class class class)co 6073    C_cat cssc 13997
This theorem is referenced by:  ssctr  14015  ssceq  14016  subcss1  14029  issubc3  14036  subsubc  14040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-ixp 7056  df-ssc 14000
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