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Theorem ressressg 12940
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 2206 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( W ↾s A ) = ( W ↾s A ) )
2 eqidd 2206 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` W ) = ( Base ` W ) )
3 simp3 1002 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> W e. Z )
4 simp1 1000 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> A e. X )
51, 2, 3, 4ressbasd 12932 . . . . . 6 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
65ineq2d 3374 . . . . 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 3383 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
8 incom 3365 . . . . . . 7 |- ( B i^i A ) = ( A i^i B )
98ineq1i 3370 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
107, 9eqtr3i 2228 . . . . 5 |- ( B i^i ( A i^i ( Base ` W ) ) ) = ( ( A i^i B ) i^i ( Base ` W ) )
116, 10eqtr3di 2253 . . . 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 3826 . . 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 5962 . 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 12930 . . . . 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 1001 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> B e. Y )
17 ressvalsets 12929 . . . 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 12929 . . . . 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 5961 . . 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 12921 . . . . 5 |- ( Base ` ndx ) e. NN
2322a1i 9 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` ndx ) e. NN )
24 inex1g 4181 . . . . 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 4181 . . . . 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 12904 . . 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 2242 . 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 4181 . . . 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 12929 . . 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 2248 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 981   = wceq 1373   e. wcel 2176   _Vcvv 2772   i^i cin 3165   <.cop 3636   ` cfv 5272  (class class class)co 5946   NNcn 9038   ndxcnx 12862   sSet csts 12863   Basecbs 12865   ↾s cress 12866
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-iota 5233  df-fun 5274  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-inn 9039  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-iress 12873
This theorem is referenced by:  ressabsg  12941
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