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Theorem ressressg 12513
Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
ressressg  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )

Proof of Theorem ressressg
StepHypRef Expression
1 eqidd 2178 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( Ws  A )  =  ( Ws  A ) )
2 eqidd 2178 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( Base `  W
)  =  ( Base `  W ) )
3 simp3 999 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  W  e.  Z )
4 simp1 997 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  A  e.  X )
51, 2, 3, 4ressbasd 12506 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( A  i^i  ( Base `  W ) )  =  ( Base `  ( Ws  A ) ) )
65ineq2d 3336 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( B  i^i  ( A  i^i  ( Base `  W
) ) )  =  ( B  i^i  ( Base `  ( Ws  A ) ) ) )
7 inass 3345 . . . . . 6  |-  ( ( B  i^i  A )  i^i  ( Base `  W
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
8 incom 3327 . . . . . . 7  |-  ( B  i^i  A )  =  ( A  i^i  B
)
98ineq1i 3332 . . . . . 6  |-  ( ( B  i^i  A )  i^i  ( Base `  W
) )  =  ( ( A  i^i  B
)  i^i  ( Base `  W ) )
107, 9eqtr3i 2200 . . . . 5  |-  ( B  i^i  ( A  i^i  ( Base `  W )
) )  =  ( ( A  i^i  B
)  i^i  ( Base `  W ) )
116, 10eqtr3di 2225 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( B  i^i  ( Base `  ( Ws  A ) ) )  =  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) )
1211opeq2d 3783 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  -> 
<. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>.  =  <. ( Base `  ndx ) ,  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) >.
)
1312oveq2d 5885 . 2  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( W sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W ) ) >.
) )
14 ressex 12504 . . . . 5  |-  ( ( W  e.  Z  /\  A  e.  X )  ->  ( Ws  A )  e.  _V )
153, 4, 14syl2anc 411 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( Ws  A )  e.  _V )
16 simp2 998 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  B  e.  Y )
17 ressvalsets 12503 . . . 4  |-  ( ( ( Ws  A )  e.  _V  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( ( Ws  A ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) ) >. )
)
1815, 16, 17syl2anc 411 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( ( Ws  A )s  B )  =  ( ( Ws  A ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) ) >. )
)
19 ressvalsets 12503 . . . . 5  |-  ( ( W  e.  Z  /\  A  e.  X )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
203, 4, 19syl2anc 411 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
2120oveq1d 5884 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( ( Ws  A ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>. )  =  (
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W ) ) >.
) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>. ) )
22 basendxnn 12497 . . . . 5  |-  ( Base `  ndx )  e.  NN
2322a1i 9 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( Base `  ndx )  e.  NN )
24 inex1g 4136 . . . . 5  |-  ( A  e.  X  ->  ( A  i^i  ( Base `  W
) )  e.  _V )
254, 24syl 14 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( A  i^i  ( Base `  W ) )  e.  _V )
26 inex1g 4136 . . . . 5  |-  ( B  e.  Y  ->  ( B  i^i  ( Base `  ( Ws  A ) ) )  e.  _V )
2716, 26syl 14 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( B  i^i  ( Base `  ( Ws  A ) ) )  e.  _V )
283, 23, 25, 27setsabsd 12481 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>. )  =  ( W sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>. ) )
2918, 21, 283eqtrd 2214 . 2  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( ( Ws  A )s  B )  =  ( W sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>. ) )
30 inex1g 4136 . . . 4  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
314, 30syl 14 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( A  i^i  B
)  e.  _V )
32 ressvalsets 12503 . . 3  |-  ( ( W  e.  Z  /\  ( A  i^i  B )  e.  _V )  -> 
( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
333, 31, 32syl2anc 411 . 2  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
3413, 29, 333eqtr4d 2220 1  |-  ( ( A  e.  X  /\  B  e.  Y  /\  W  e.  Z )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128   <.cop 3594   ` cfv 5212  (class class class)co 5869   NNcn 8905   ndxcnx 12439   sSet csts 12440   Basecbs 12442   ↾s cress 12443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-inn 8906  df-ndx 12445  df-slot 12446  df-base 12448  df-sets 12449  df-iress 12450
This theorem is referenced by:  ressabsg  12514
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