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Theorem ssc2 14022
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 2437 . . . . . 6  |-  ( ph  ->  dom  dom  J  =  dom  dom  J )
63, 5sscfn2 14018 . . . . 5  |-  ( ph  ->  J  Fn  ( dom 
dom  J  X.  dom  dom  J ) )
7 sscrel 14013 . . . . . . . 8  |-  Rel  C_cat
87brrelex2i 4919 . . . . . . 7  |-  ( H 
C_cat  J  ->  J  e.  _V )
93, 8syl 16 . . . . . 6  |-  ( ph  ->  J  e.  _V )
10 dmexg 5130 . . . . . 6  |-  ( J  e.  _V  ->  dom  J  e.  _V )
11 dmexg 5130 . . . . . 6  |-  ( dom 
J  e.  _V  ->  dom 
dom  J  e.  _V )
129, 10, 113syl 19 . . . . 5  |-  ( ph  ->  dom  dom  J  e.  _V )
134, 6, 12isssc 14020 . . . 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 ) ) ) )
143, 13mpbid 202 . . 3  |-  ( ph  ->  ( S  C_  dom  dom 
J  /\  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) ) )
1514simprd 450 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x H y )  C_  ( x J y ) )
16 oveq1 6088 . . . 4  |-  ( x  =  X  ->  (
x H y )  =  ( X H y ) )
17 oveq1 6088 . . . 4  |-  ( x  =  X  ->  (
x J y )  =  ( X J y ) )
1816, 17sseq12d 3377 . . 3  |-  ( x  =  X  ->  (
( x H y )  C_  ( x J y )  <->  ( X H y )  C_  ( X J y ) ) )
19 oveq2 6089 . . . 4  |-  ( y  =  Y  ->  ( X H y )  =  ( X H Y ) )
20 oveq2 6089 . . . 4  |-  ( y  =  Y  ->  ( X J y )  =  ( X J Y ) )
2119, 20sseq12d 3377 . . 3  |-  ( y  =  Y  ->  (
( X H y )  C_  ( X J y )  <->  ( X H Y )  C_  ( X J Y ) ) )
2218, 21rspc2va 3059 . 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 ) )
231, 2, 15, 22syl21anc 1183 1  |-  ( ph  ->  ( X H Y )  C_  ( X J Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    C_ wss 3320   class class class wbr 4212    X. cxp 4876   dom cdm 4878    Fn wfn 5449  (class class class)co 6081    C_cat cssc 14007
This theorem is referenced by:  ssctr  14025  ssceq  14026  subcss2  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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-ixp 7064  df-ssc 14010
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