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Theorem ressinbas 14794
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 3065 . 2  |-  ( A  e.  X  ->  A  e.  _V )
2 eqid 2400 . . . . . . 7  |-  ( Ws  A )  =  ( Ws  A )
3 ressid.1 . . . . . . 7  |-  B  =  ( Base `  W
)
42, 3ressid2 14786 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  W )
5 ssid 3458 . . . . . . . 8  |-  B  C_  B
6 incom 3629 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
7 df-ss 3425 . . . . . . . . . 10  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
87biimpi 194 . . . . . . . . 9  |-  ( B 
C_  A  ->  ( B  i^i  A )  =  B )
96, 8syl5eq 2453 . . . . . . . 8  |-  ( B 
C_  A  ->  ( A  i^i  B )  =  B )
105, 9syl5sseqr 3488 . . . . . . 7  |-  ( B 
C_  A  ->  B  C_  ( A  i^i  B
) )
11 elex 3065 . . . . . . 7  |-  ( W  e.  _V  ->  W  e.  _V )
12 inex1g 4534 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  i^i  B )  e. 
_V )
13 eqid 2400 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
1413, 3ressid2 14786 . . . . . . 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 2444 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
17163expb 1196 . . . 4  |-  ( ( B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
18 inass 3646 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  B )  =  ( A  i^i  ( B  i^i  B ) )
19 inidm 3645 . . . . . . . . . 10  |-  ( B  i^i  B )  =  B
2019ineq2i 3635 . . . . . . . . 9  |-  ( A  i^i  ( B  i^i  B ) )  =  ( A  i^i  B )
2118, 20eqtr2i 2430 . . . . . . . 8  |-  ( A  i^i  B )  =  ( ( A  i^i  B )  i^i  B )
2221opeq2i 4160 . . . . . . 7  |-  <. ( Base `  ndx ) ,  ( A  i^i  B
) >.  =  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>.
2322oveq2i 6243 . . . . . 6  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>. )
242, 3ressval2 14787 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
25 inss1 3656 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
26 sstr 3447 . . . . . . . . 9  |-  ( ( B  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  B  C_  A )
2725, 26mpan2 669 . . . . . . . 8  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
2827con3i 135 . . . . . . 7  |-  ( -.  B  C_  A  ->  -.  B  C_  ( A  i^i  B ) )
2913, 3ressval2 14787 . . . . . . 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 2467 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
32313expb 1196 . . . 4  |-  ( ( -.  B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
3317, 32pm2.61ian 789 . . 3  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
34 reldmress 14784 . . . . . 6  |-  Rel  doms
3534ovprc1 6263 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
3634ovprc1 6263 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
3735, 36eqtr4d 2444 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3837adantr 463 . . 3  |-  ( ( -.  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3933, 38pm2.61ian 789 . 2  |-  ( A  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
401, 39syl 17 1  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   _Vcvv 3056    i^i cin 3410    C_ wss 3411   (/)c0 3735   <.cop 3975   ` cfv 5523  (class class class)co 6232   ndxcnx 14728   sSet csts 14729   Basecbs 14731   ↾s cress 14732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-ress 14738
This theorem is referenced by:  ressress  14796  rescabs  15336  resscat  15355  funcres2c  15404  ressffth  15441  cphsubrglem  21806  suborng  28139
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