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Theorem ressressg 13305
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 2235 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( W ↾s A ) = ( W ↾s A ) )
2 eqidd 2235 . . . . . . 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 13297 . . . . . 6 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
65ineq2d 3424 . . . . 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 3433 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
8 incom 3413 . . . . . . 7 |- ( B i^i A ) = ( A i^i B )
98ineq1i 3420 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
107, 9eqtr3i 2257 . . . . 5 |- ( B i^i ( A i^i ( Base ` W ) ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
116, 10eqtr3di 2282 . . . 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 3892 . . 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 6068 . 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 13295 . . . . 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 13294 . . . 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 13294 . . . . 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 6067 . . 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 13285 . . . . 5 |- ( Base ` ndx ) e. NN
2322a1i 9 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` ndx ) e. NN )
24 inex1g 4248 . . . . 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 4248 . . . . 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 13268 . . 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 2271 . 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 4248 . . . 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 13294 . . 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 2277 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 2205   _Vcvv 2815   i^i cin 3212   <.cop 3694   ` cfv 5354  (class class class)co 6052   NNcn 9239   ndxcnx 13226   sSet csts 13227   Basecbs 13229   ↾s cress 13230
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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-inn 9240  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237
This theorem is referenced by:  ressabsg  13306
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