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Theorem ressinbas 14567
Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressinbas  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )

Proof of Theorem ressinbas
StepHypRef Expression
1 elex 3127 . 2  |-  ( A  e.  X  ->  A  e.  _V )
2 eqid 2467 . . . . . . 7  |-  ( Ws  A )  =  ( Ws  A )
3 ressid.1 . . . . . . 7  |-  B  =  ( Base `  W
)
42, 3ressid2 14559 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  W )
5 ssid 3528 . . . . . . . 8  |-  B  C_  B
6 incom 3696 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
7 df-ss 3495 . . . . . . . . . 10  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
87biimpi 194 . . . . . . . . 9  |-  ( B 
C_  A  ->  ( B  i^i  A )  =  B )
96, 8syl5eq 2520 . . . . . . . 8  |-  ( B 
C_  A  ->  ( A  i^i  B )  =  B )
105, 9syl5sseqr 3558 . . . . . . 7  |-  ( B 
C_  A  ->  B  C_  ( A  i^i  B
) )
11 elex 3127 . . . . . . 7  |-  ( W  e.  _V  ->  W  e.  _V )
12 inex1g 4596 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  i^i  B )  e. 
_V )
13 eqid 2467 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
1413, 3ressid2 14559 . . . . . . 7  |-  ( ( B  C_  ( A  i^i  B )  /\  W  e.  _V  /\  ( A  i^i  B )  e. 
_V )  ->  ( Ws  ( A  i^i  B ) )  =  W )
1510, 11, 12, 14syl3an 1270 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  ( A  i^i  B ) )  =  W )
164, 15eqtr4d 2511 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
17163expb 1197 . . . 4  |-  ( ( B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
18 inass 3713 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  B )  =  ( A  i^i  ( B  i^i  B ) )
19 inidm 3712 . . . . . . . . . 10  |-  ( B  i^i  B )  =  B
2019ineq2i 3702 . . . . . . . . 9  |-  ( A  i^i  ( B  i^i  B ) )  =  ( A  i^i  B )
2118, 20eqtr2i 2497 . . . . . . . 8  |-  ( A  i^i  B )  =  ( ( A  i^i  B )  i^i  B )
2221opeq2i 4223 . . . . . . 7  |-  <. ( Base `  ndx ) ,  ( A  i^i  B
) >.  =  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>.
2322oveq2i 6306 . . . . . 6  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>. )
242, 3ressval2 14560 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
25 inss1 3723 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
26 sstr 3517 . . . . . . . . 9  |-  ( ( B  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  B  C_  A )
2725, 26mpan2 671 . . . . . . . 8  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
2827con3i 135 . . . . . . 7  |-  ( -.  B  C_  A  ->  -.  B  C_  ( A  i^i  B ) )
2913, 3ressval2 14560 . . . . . . 7  |-  ( ( -.  B  C_  ( A  i^i  B )  /\  W  e.  _V  /\  ( A  i^i  B )  e. 
_V )  ->  ( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i 
B ) >. )
)
3028, 11, 12, 29syl3an 1270 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i 
B ) >. )
)
3123, 24, 303eqtr4a 2534 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
32313expb 1197 . . . 4  |-  ( ( -.  B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
3317, 32pm2.61ian 788 . . 3  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
34 reldmress 14557 . . . . . 6  |-  Rel  doms
3534ovprc1 6323 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
3634ovprc1 6323 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
3735, 36eqtr4d 2511 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3837adantr 465 . . 3  |-  ( ( -.  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3933, 38pm2.61ian 788 . 2  |-  ( A  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
401, 39syl 16 1  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   <.cop 4039   ` cfv 5594  (class class class)co 6295   ndxcnx 14503   sSet csts 14504   Basecbs 14506   ↾s cress 14507
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-ress 14513
This theorem is referenced by:  ressress  14568  rescabs  15079  resscat  15095  funcres2c  15144  ressffth  15181  cphsubrglem  21490  suborng  27637
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