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Theorem ssc2 15310
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 2455 . . . . . 6  |-  ( ph  ->  dom  dom  J  =  dom  dom  J )
63, 5sscfn2 15306 . . . . 5  |-  ( ph  ->  J  Fn  ( dom 
dom  J  X.  dom  dom  J ) )
7 sscrel 15301 . . . . . . 7  |-  Rel  C_cat
87brrelex2i 5030 . . . . . 6  |-  ( H 
C_cat  J  ->  J  e.  _V )
9 dmexg 6704 . . . . . 6  |-  ( J  e.  _V  ->  dom  J  e.  _V )
10 dmexg 6704 . . . . . 6  |-  ( dom 
J  e.  _V  ->  dom 
dom  J  e.  _V )
113, 8, 9, 104syl 21 . . . . 5  |-  ( ph  ->  dom  dom  J  e.  _V )
124, 6, 11isssc 15308 . . . 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 ) ) ) )
133, 12mpbid 210 . . 3  |-  ( ph  ->  ( S  C_  dom  dom 
J  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) )
1413simprd 461 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) )
15 oveq1 6277 . . . 4  |-  ( x  =  X  ->  (
x H y )  =  ( X H y ) )
16 oveq1 6277 . . . 4  |-  ( x  =  X  ->  (
x J y )  =  ( X J y ) )
1715, 16sseq12d 3518 . . 3  |-  ( x  =  X  ->  (
( x H y )  C_  ( x J y )  <->  ( X H y )  C_  ( X J y ) ) )
18 oveq2 6278 . . . 4  |-  ( y  =  Y  ->  ( X H y )  =  ( X H Y ) )
19 oveq2 6278 . . . 4  |-  ( y  =  Y  ->  ( X J y )  =  ( X J Y ) )
2018, 19sseq12d 3518 . . 3  |-  ( y  =  Y  ->  (
( X H y )  C_  ( X J y )  <->  ( X H Y )  C_  ( X J Y ) ) )
2117, 20rspc2va 3217 . 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 ) )
221, 2, 14, 21syl21anc 1225 1  |-  ( ph  ->  ( X H Y )  C_  ( X J Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    X. cxp 4986   dom cdm 4988    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  subcss2  15331
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