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Theorem sscres 13700
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 3389 . . 3  |-  ( S  i^i  T )  C_  S
2 inss2 3390 . . . . . . 7  |-  ( S  i^i  T )  C_  T
3 simpl 443 . . . . . . 7  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  ->  x  e.  ( S  i^i  T ) )
42, 3sseldi 3178 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  ->  x  e.  T )
5 simpr 447 . . . . . . 7  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  ( S  i^i  T ) )
62, 5sseldi 3178 . . . . . 6  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
y  e.  T )
74, 6ovresd 5988 . . . . 5  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
( x ( H  |`  ( T  X.  T
) ) y )  =  ( x H y ) )
8 eqimss 3230 . . . . 5  |-  ( ( x ( H  |`  ( T  X.  T
) ) y )  =  ( x H y )  ->  (
x ( H  |`  ( T  X.  T
) ) y ) 
C_  ( x H y ) )
97, 8syl 15 . . . 4  |-  ( ( x  e.  ( S  i^i  T )  /\  y  e.  ( S  i^i  T ) )  -> 
( x ( H  |`  ( T  X.  T
) ) y ) 
C_  ( x H y ) )
109rgen2 2639 . . 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 441 . 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 443 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  H  Fn  ( S  X.  S ) )
13 inss1 3389 . . . . 5  |-  ( ( S  X.  S )  i^i  ( T  X.  T ) )  C_  ( S  X.  S
)
14 fnssres 5357 . . . . 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 643 . . . 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 4968 . . . . . 6  |-  ( ( H  |`  ( S  X.  S ) )  |`  ( T  X.  T
) )  =  ( H  |`  ( ( S  X.  S )  i^i  ( T  X.  T
) ) )
17 fnresdm 5353 . . . . . . . 8  |-  ( H  Fn  ( S  X.  S )  ->  ( H  |`  ( S  X.  S ) )  =  H )
1817adantr 451 . . . . . . 7  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  ( S  X.  S ) )  =  H )
1918reseq1d 4954 . . . . . 6  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( ( H  |`  ( S  X.  S
) )  |`  ( T  X.  T ) )  =  ( H  |`  ( T  X.  T
) ) )
2016, 19syl5eqr 2329 . . . . 5  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  ( H  |`  (
( S  X.  S
)  i^i  ( T  X.  T ) ) )  =  ( H  |`  ( T  X.  T
) ) )
21 inxp 4818 . . . . . 6  |-  ( ( S  X.  S )  i^i  ( T  X.  T ) )  =  ( ( S  i^i  T )  X.  ( S  i^i  T ) )
2221a1i 10 . . . . 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 5337 . . . 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 201 . . 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 447 . . 3  |-  ( ( H  Fn  ( S  X.  S )  /\  S  e.  V )  ->  S  e.  V )
2624, 12, 25isssc 13697 . 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 224 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 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023    X. cxp 4687    |` cres 4691    Fn wfn 5250  (class class class)co 5858    C_cat cssc 13684
This theorem is referenced by:  sscid  13701  fullsubc  13724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-ixp 6818  df-ssc 13687
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