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Theorem sscres 14847
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 3671 . . 3  |-  ( S  i^i  T )  C_  S
2 inss2 3672 . . . . . . 7  |-  ( S  i^i  T )  C_  T
3 simpl 457 . . . . . . 7  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  ->  x  e.  ( S  i^i  T ) )
42, 3sseldi 3455 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  ->  x  e.  T )
5 simpr 461 . . . . . . 7  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  ( S  i^i  T ) )
62, 5sseldi 3455 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  T )
74, 6ovresd 6334 . . . . 5  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
( x ( H  |`  ( T  X.  T
) ) y )  =  ( x H y ) )
8 eqimss 3509 . . . . 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 2893 . . 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 455 . 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 457 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  H  Fn  ( S  X.  S ) )
13 inss1 3671 . . . . 5  |-  ( ( S  X.  S )  i^i  ( T  X.  T ) )  C_  ( S  X.  S
)
14 fnssres 5625 . . . . 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 662 . . . 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 5224 . . . . . 6  |-  ( ( H  |`  ( S  X.  S ) )  |`  ( T  X.  T
) )  =  ( H  |`  ( ( S  X.  S )  i^i  ( T  X.  T
) ) )
17 fnresdm 5621 . . . . . . . 8  |-  ( H  Fn  ( S  X.  S )  ->  ( H  |`  ( S  X.  S ) )  =  H )
1817adantr 465 . . . . . . 7  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  ( S  X.  S ) )  =  H )
1918reseq1d 5210 . . . . . 6  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( H  |`  ( S  X.  S
) )  |`  ( T  X.  T ) )  =  ( H  |`  ( T  X.  T
) ) )
2016, 19syl5eqr 2506 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  (
( S  X.  S
)  i^i  ( T  X.  T ) ) )  =  ( H  |`  ( T  X.  T
) ) )
21 inxp 5073 . . . . . 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 5604 . . . 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 210 . . 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 461 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  S  e.  V )
2624, 12, 25isssc 14844 . 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 233 1  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  ( T  X.  T ) ) 
C_cat  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    i^i cin 3428    C_ wss 3429   class class class wbr 4393    X. cxp 4939    |` cres 4943    Fn wfn 5514  (class class class)co 6193    C_cat cssc 14831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-ixp 7367  df-ssc 14834
This theorem is referenced by:  sscid  14848  fullsubc  14871
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