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Theorem ressressg 13238
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 2232 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( W ↾s A ) = ( W ↾s A ) )
2 eqidd 2232 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` W ) = ( Base ` W ) )
3 simp3 1026 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> W e. Z )
4 simp1 1024 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> A e. X )
51, 2, 3, 4ressbasd 13230 . . . . . 6 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
65ineq2d 3410 . . . . 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 3419 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
8 incom 3401 . . . . . . 7 |- ( B i^i A ) = ( A i^i B )
98ineq1i 3406 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
107, 9eqtr3i 2254 . . . . 5 |- ( B i^i ( A i^i ( Base ` W ) ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
116, 10eqtr3di 2279 . . . 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 3874 . . 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 6044 . 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 13228 . . . . 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 1025 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> B e. Y )
17 ressvalsets 13227 . . . 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 13227 . . . . 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 6043 . . 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 13218 . . . . 5 |- ( Base ` ndx ) e. NN
2322a1i 9 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` ndx ) e. NN )
24 inex1g 4230 . . . . 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 4230 . . . . 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 13201 . . 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 2268 . 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 4230 . . . 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 13227 . . 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 2274 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 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   <.cop 3676   ` cfv 5333  (class class class)co 6028   NNcn 9202   ndxcnx 13159   sSet csts 13160   Basecbs 13162   ↾s cress 13163
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9203  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170
This theorem is referenced by:  ressabsg  13239
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