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Theorem sscres 15049
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 3718 . . 3  |-  ( S  i^i  T )  C_  S
2 inss2 3719 . . . . . . 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 3502 . . . . . 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 3502 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  T )
74, 6ovresd 6425 . . . . 5  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
( x ( H  |`  ( T  X.  T
) ) y )  =  ( x H y ) )
8 eqimss 3556 . . . . 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 2891 . . 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 3718 . . . . 5  |-  ( ( S  X.  S )  i^i  ( T  X.  T ) )  C_  ( S  X.  S
)
14 fnssres 5692 . . . . 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 5284 . . . . . 6  |-  ( ( H  |`  ( S  X.  S ) )  |`  ( T  X.  T
) )  =  ( H  |`  ( ( S  X.  S )  i^i  ( T  X.  T
) ) )
17 fnresdm 5688 . . . . . . . 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 5270 . . . . . 6  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( H  |`  ( S  X.  S
) )  |`  ( T  X.  T ) )  =  ( H  |`  ( T  X.  T
) ) )
2016, 19syl5eqr 2522 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  (
( S  X.  S
)  i^i  ( T  X.  T ) ) )  =  ( H  |`  ( T  X.  T
) ) )
21 inxp 5133 . . . . . 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 5671 . . . 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 15046 . 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 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476   class class class wbr 4447    X. cxp 4997    |` cres 5001    Fn wfn 5581  (class class class)co 6282    C_cat cssc 15033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-ixp 7467  df-ssc 15036
This theorem is referenced by:  sscid  15050  fullsubc  15073
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