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Theorem sscres 13950
Description: Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
sscres  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  ( T  X.  T ) ) 
C_cat  H )

Proof of Theorem sscres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3504 . . 3  |-  ( S  i^i  T )  C_  S
2 inss2 3505 . . . . . . 7  |-  ( S  i^i  T )  C_  T
3 simpl 444 . . . . . . 7  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  ->  x  e.  ( S  i^i  T ) )
42, 3sseldi 3289 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  ->  x  e.  T )
5 simpr 448 . . . . . . 7  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  ( S  i^i  T ) )
62, 5sseldi 3289 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  T )
74, 6ovresd 6153 . . . . 5  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
( x ( H  |`  ( T  X.  T
) ) y )  =  ( x H y ) )
8 eqimss 3343 . . . . 5  |-  ( ( x ( H  |`  ( T  X.  T
) ) y )  =  ( x H y )  ->  (
x ( H  |`  ( T  X.  T
) ) y ) 
C_  ( x H y ) )
97, 8syl 16 . . . 4  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
( x ( H  |`  ( T  X.  T
) ) y ) 
C_  ( x H y ) )
109rgen2a 2715 . . 3  |-  A. x  e.  ( S  i^i  T
) A. y  e.  ( S  i^i  T
) ( x ( H  |`  ( T  X.  T ) ) y )  C_  ( x H y )
111, 10pm3.2i 442 . 2  |-  ( ( S  i^i  T ) 
C_  S  /\  A. x  e.  ( S  i^i  T ) A. y  e.  ( S  i^i  T
) ( x ( H  |`  ( T  X.  T ) ) y )  C_  ( x H y ) )
12 simpl 444 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  H  Fn  ( S  X.  S ) )
13 inss1 3504 . . . . 5  |-  ( ( S  X.  S )  i^i  ( T  X.  T ) )  C_  ( S  X.  S
)
14 fnssres 5498 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  ( ( S  X.  S )  i^i  ( T  X.  T ) ) 
C_  ( S  X.  S ) )  -> 
( H  |`  (
( S  X.  S
)  i^i  ( T  X.  T ) ) )  Fn  ( ( S  X.  S )  i^i  ( T  X.  T
) ) )
1512, 13, 14sylancl 644 . . . 4  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  (
( S  X.  S
)  i^i  ( T  X.  T ) ) )  Fn  ( ( S  X.  S )  i^i  ( T  X.  T
) ) )
16 resres 5099 . . . . . 6  |-  ( ( H  |`  ( S  X.  S ) )  |`  ( T  X.  T
) )  =  ( H  |`  ( ( S  X.  S )  i^i  ( T  X.  T
) ) )
17 fnresdm 5494 . . . . . . . 8  |-  ( H  Fn  ( S  X.  S )  ->  ( H  |`  ( S  X.  S ) )  =  H )
1817adantr 452 . . . . . . 7  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  ( S  X.  S ) )  =  H )
1918reseq1d 5085 . . . . . 6  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( H  |`  ( S  X.  S
) )  |`  ( T  X.  T ) )  =  ( H  |`  ( T  X.  T
) ) )
2016, 19syl5eqr 2433 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  (
( S  X.  S
)  i^i  ( T  X.  T ) ) )  =  ( H  |`  ( T  X.  T
) ) )
21 inxp 4947 . . . . . 6  |-  ( ( S  X.  S )  i^i  ( T  X.  T ) )  =  ( ( S  i^i  T )  X.  ( S  i^i  T ) )
2221a1i 11 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( S  X.  S )  i^i  ( T  X.  T ) )  =  ( ( S  i^i  T )  X.  ( S  i^i  T
) ) )
2320, 22fneq12d 5478 . . . 4  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( H  |`  ( ( S  X.  S )  i^i  ( T  X.  T ) ) )  Fn  ( ( S  X.  S )  i^i  ( T  X.  T ) )  <->  ( H  |`  ( T  X.  T
) )  Fn  (
( S  i^i  T
)  X.  ( S  i^i  T ) ) ) )
2415, 23mpbid 202 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  ( T  X.  T ) )  Fn  ( ( S  i^i  T )  X.  ( S  i^i  T
) ) )
25 simpr 448 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  S  e.  V )
2624, 12, 25isssc 13947 . 2  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( H  |`  ( T  X.  T
) )  C_cat  H  <->  ( ( S  i^i  T )  C_  S  /\  A. x  e.  ( S  i^i  T
) A. y  e.  ( S  i^i  T
) ( x ( H  |`  ( T  X.  T ) ) y )  C_  ( x H y ) ) ) )
2711, 26mpbiri 225 1  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  ( T  X.  T ) ) 
C_cat  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    i^i cin 3262    C_ wss 3263   class class class wbr 4153    X. cxp 4816    |` cres 4820    Fn wfn 5389  (class class class)co 6020    C_cat cssc 13934
This theorem is referenced by:  sscid  13951  fullsubc  13974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-ixp 7000  df-ssc 13937
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