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Theorem ressinbas 13204
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 2796 . 2  |-  ( A  e.  X  ->  A  e.  _V )
2 eqid 2283 . . . . . . 7  |-  ( Ws  A )  =  ( Ws  A )
3 ressid.1 . . . . . . 7  |-  B  =  ( Base `  W
)
42, 3ressid2 13196 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  W )
5 ssid 3197 . . . . . . . 8  |-  B  C_  B
6 incom 3361 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
7 df-ss 3166 . . . . . . . . . 10  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
87biimpi 186 . . . . . . . . 9  |-  ( B 
C_  A  ->  ( B  i^i  A )  =  B )
96, 8syl5eq 2327 . . . . . . . 8  |-  ( B 
C_  A  ->  ( A  i^i  B )  =  B )
105, 9syl5sseqr 3227 . . . . . . 7  |-  ( B 
C_  A  ->  B  C_  ( A  i^i  B
) )
11 elex 2796 . . . . . . 7  |-  ( W  e.  _V  ->  W  e.  _V )
12 inex1g 4157 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  i^i  B )  e. 
_V )
13 eqid 2283 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
1413, 3ressid2 13196 . . . . . . 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 1224 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  ( A  i^i  B ) )  =  W )
164, 15eqtr4d 2318 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
17163expb 1152 . . . 4  |-  ( ( B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
18 inass 3379 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  B )  =  ( A  i^i  ( B  i^i  B ) )
19 inidm 3378 . . . . . . . . . 10  |-  ( B  i^i  B )  =  B
2019ineq2i 3367 . . . . . . . . 9  |-  ( A  i^i  ( B  i^i  B ) )  =  ( A  i^i  B )
2118, 20eqtr2i 2304 . . . . . . . 8  |-  ( A  i^i  B )  =  ( ( A  i^i  B )  i^i  B )
2221opeq2i 3800 . . . . . . 7  |-  <. ( Base `  ndx ) ,  ( A  i^i  B
) >.  =  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>.
2322oveq2i 5869 . . . . . 6  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>. )
242, 3ressval2 13197 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
25 inss1 3389 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
26 sstr 3187 . . . . . . . . 9  |-  ( ( B  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  B  C_  A )
2725, 26mpan2 652 . . . . . . . 8  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
2827con3i 127 . . . . . . 7  |-  ( -.  B  C_  A  ->  -.  B  C_  ( A  i^i  B ) )
2913, 3ressval2 13197 . . . . . . 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 1224 . . . . . 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 2341 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
32313expb 1152 . . . 4  |-  ( ( -.  B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
3317, 32pm2.61ian 765 . . 3  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
34 reldmress 13194 . . . . . 6  |-  Rel  doms
3534ovprc1 5886 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
3634ovprc1 5886 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
3735, 36eqtr4d 2318 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3837adantr 451 . . 3  |-  ( ( -.  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3933, 38pm2.61ian 765 . 2  |-  ( A  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
401, 39syl 15 1  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   <.cop 3643   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149
This theorem is referenced by:  ressress  13205  rescabs  13710  resscat  13726  funcres2c  13775  ressffth  13812  cphsubrglem  18613
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ress 13155
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