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Theorem ssc1 15067
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 15059 . . . . . . 7  |-  Rel  C_cat
54brrelex2i 5031 . . . . . 6  |-  ( H 
C_cat  J  ->  J  e.  _V )
61, 5syl 16 . . . . 5  |-  ( ph  ->  J  e.  _V )
73ssclem 15065 . . . . 5  |-  ( ph  ->  ( J  e.  _V  <->  T  e.  _V ) )
86, 7mpbid 210 . . . 4  |-  ( ph  ->  T  e.  _V )
92, 3, 8isssc 15066 . . 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 459 1  |-  ( ph  ->  S  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1804   A.wral 2793   _Vcvv 3095    C_ wss 3461   class class class wbr 4437    X. cxp 4987    Fn wfn 5573  (class class class)co 6281    C_cat cssc 15053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-ixp 7472  df-ssc 15056
This theorem is referenced by:  ssctr  15071  ssceq  15072  subcss1  15085  issubc3  15092  subsubc  15096
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