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Theorem ressressg 12590
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 ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
Proof of Theorem ressressg
StepHypRef Expression
1 eqidd 2190 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( W ↾s A ) = ( W ↾s A ) )
2 eqidd 2190 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` W ) = ( Base ` W ) )
3 simp3 1001 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> W e. Z )
4 simp1 999 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> A e. X )
51, 2, 3, 4ressbasd 12582 . . . . . 6 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
65ineq2d 3351 . . . . 5 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( B i^i ( A i^i ( Base ` W ) ) ) = ( B i^i ( Base ` ( W ↾s A ) ) ) )
7 inass 3360 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
8 incom 3342 . . . . . . 7 |- ( B i^i A ) = ( A i^i B )
98ineq1i 3347 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
107, 9eqtr3i 2212 . . . . 5 |- ( B i^i ( A i^i ( Base ` W ) ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
116, 10eqtr3di 2237 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( B i^i ( Base ` ( W ↾s A ) ) ) = ( ( A i^i B ) i^i ( Base ` W ) ) )
1211opeq2d 3800 . . 3 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. = <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. )
1312oveq2d 5913 . 2 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( W sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
14 ressex 12580 . . . . 5 |- ( ( W e. Z /\ A e. X ) -> ( W ↾s A ) e. _V )
153, 4, 14syl2anc 411 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( W ↾s A ) e. _V )
16 simp2 1000 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> B e. Y )
17 ressvalsets 12579 . . . 4 |- ( ( ( W ↾s A ) e. _V /\ B e. Y ) -> ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) )
1815, 16, 17syl2anc 411 . . 3 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( ( W ↾s A ) ↾s B ) = ( ( W ↾s A ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) )
19 ressvalsets 12579 . . . . 5 |- ( ( W e. Z /\ A e. X ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
203, 4, 19syl2anc 411 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
2120oveq1d 5912 . . 3 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( ( W ↾s A ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) = ( ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) )
22 basendxnn 12571 . . . . 5 |- ( Base ` ndx ) e. NN
2322a1i 9 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` ndx ) e. NN )
24 inex1g 4154 . . . . 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 4154 . . . . 5 |- ( B e. Y -> ( B i^i ( Base ` ( W ↾s A ) ) ) e. _V )
2716, 26syl 14 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( B i^i ( Base ` ( W ↾s A ) ) ) e. _V )
283, 23, 25, 27setsabsd 12554 . . 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 ` ( W ↾s A ) ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) )
2918, 21, 283eqtrd 2226 . 2 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( ( W ↾s A ) ↾s B ) = ( W sSet <. ( Base ` ndx ) , ( B i^i ( Base ` ( W ↾s A ) ) ) >. ) )
30 inex1g 4154 . . . 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 12579 . . 3 |- ( ( W e. Z /\ ( A i^i B ) e. _V ) -> ( W ↾s ( 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 ) -> ( W ↾s ( A i^i B ) ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
3413, 29, 333eqtr4d 2232 1 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( ( W ↾s A ) ↾s B ) = ( W ↾s ( A i^i B ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160   _Vcvv 2752   i^i cin 3143   <.cop 3610   ` cfv 5235  (class class class)co 5897   NNcn 8950   ndxcnx 12512   sSet csts 12513   Basecbs 12515   ↾s cress 12516
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-inn 8951  df-ndx 12518  df-slot 12519  df-base 12521  df-sets 12522  df-iress 12523
This theorem is referenced by:  ressabsg  12591
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