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Theorem sscres 14728
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 3565 . . 3  |-  ( S  i^i  T )  C_  S
2 inss2 3566 . . . . . . 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 3349 . . . . . 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 3349 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  T )
74, 6ovresd 6226 . . . . 5  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
( x ( H  |`  ( T  X.  T
) ) y )  =  ( x H y ) )
8 eqimss 3403 . . . . 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 2777 . . 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 3565 . . . . 5  |-  ( ( S  X.  S )  i^i  ( T  X.  T ) )  C_  ( S  X.  S
)
14 fnssres 5519 . . . . 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 5118 . . . . . 6  |-  ( ( H  |`  ( S  X.  S ) )  |`  ( T  X.  T
) )  =  ( H  |`  ( ( S  X.  S )  i^i  ( T  X.  T
) ) )
17 fnresdm 5515 . . . . . . . 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 5104 . . . . . 6  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( H  |`  ( S  X.  S
) )  |`  ( T  X.  T ) )  =  ( H  |`  ( T  X.  T
) ) )
2016, 19syl5eqr 2484 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  (
( S  X.  S
)  i^i  ( T  X.  T ) ) )  =  ( H  |`  ( T  X.  T
) ) )
21 inxp 4967 . . . . . 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 5498 . . . 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 14725 . 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 1369    e. wcel 1756   A.wral 2710    i^i cin 3322    C_ wss 3323   class class class wbr 4287    X. cxp 4833    |` cres 4837    Fn wfn 5408  (class class class)co 6086    C_cat cssc 14712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-ixp 7256  df-ssc 14715
This theorem is referenced by:  sscid  14729  fullsubc  14752
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