Theorem ressinbasd 12906

Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Hypotheses

Ref	Expression
ressidbasd.1	|- ( ph -> B = ( Base ` W ) )
ressidbasd.a	|- ( ph -> A e. X )
ressidbasd.w	|- ( ph -> W e. V )

Assertion

Ref	Expression
ressinbasd	|- ( ph -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )

Proof of Theorem ressinbasd

Step	Hyp	Ref	Expression
1		ressidbasd.1	. . . . . . 7 |- ( ph -> B = ( Base ` W ) )
2		inidm 3382	. . . . . . . 8 |- ( B i^i B ) = B
3	1	ineq2d 3374	. . . . . . . 8 |- ( ph -> ( B i^i B ) = ( B i^i ( Base ` W ) ) )
4	2, 3	eqtr3id 2252	. . . . . . 7 |- ( ph -> B = ( B i^i ( Base ` W ) ) )
5	1, 4	eqtr3d 2240	. . . . . 6 |- ( ph -> ( Base ` W ) = ( B i^i ( Base ` W ) ) )
6	5	ineq2d 3374	. . . . 5 |- ( ph -> ( A i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) ) )
7		inass 3383	. . . . 5 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
8	6, 7	eqtr4di 2256	. . . 4 |- ( ph -> ( A i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) ) )
9	8	opeq2d 3826	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. )
10	9	oveq2d 5960	. 2 |- ( ph -> ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
11		ressidbasd.w	. . 3 |- ( ph -> W e. V )
12		ressidbasd.a	. . 3 |- ( ph -> A e. X )
13		ressvalsets 12896	. . 3 |- ( ( W e. V /\ A e. X ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
14	11, 12, 13	syl2anc 411	. 2 |- ( ph -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
15		inex1g 4180	. . . 4 |- ( A e. X -> ( A i^i B ) e. _V )
16	12, 15	syl 14	. . 3 |- ( ph -> ( A i^i B ) e. _V )
17		ressvalsets 12896	. . 3 |- ( ( W e. V /\ ( A i^i B ) e. _V ) -> ( W ↾s ( A i^i B ) ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
18	11, 16, 17	syl2anc 411	. 2 |- ( ph -> ( W ↾s ( A i^i B ) ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. ) )
19	10, 14, 18	3eqtr4d 2248	1 |- ( ph -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   i^i cin 3165   <.cop 3636   ` cfv 5271  (class class class)co 5944   ndxcnx 12829   sSet csts 12830   Basecbs 12832   ↾_s cress 12833

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 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

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-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840

This theorem is referenced by: (None)