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Theorem ressressg 13157
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 1025 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> W e. Z )
4 simp1 1023 . . . . . . 7 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> A e. X )
51, 2, 3, 4ressbasd 13149 . . . . . 6 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W ↾s A ) ) )
65ineq2d 3408 . . . . 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 3417 . . . . . 6 |- ( ( B i^i A ) i^i ( Base ` W ) ) = ( B i^i ( A i^i ( Base ` W ) ) )
8 incom 3399 . . . . . . 7 |- ( B i^i A ) = ( A i^i B )
98ineq1i 3404 . . . . . 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 3869 . . 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 6033 . 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 13147 . . . . 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 1024 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> B e. Y )
17 ressvalsets 13146 . . . 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 13146 . . . . 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 6032 . . 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 13137 . . . . 5 |- ( Base ` ndx ) e. NN
2322a1i 9 . . . 4 |- ( ( A e. X /\ B e. Y /\ W e. Z ) -> ( Base ` ndx ) e. NN )
24 inex1g 4225 . . . . 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 4225 . . . . 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 13120 . . 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 4225 . . . 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 13146 . . 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 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   i^i cin 3199   <.cop 3672   ` cfv 5326  (class class class)co 6017   NNcn 9142   ndxcnx 13078   sSet csts 13079   Basecbs 13081   ↾s cress 13082
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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089
This theorem is referenced by:  ressabsg  13158
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