Theorem ressinbasd 12777

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 3373	. . . . . . . 8 |- ( B i^i B ) = B
3	1	ineq2d 3365	. . . . . . . 8 |- ( ph -> ( B i^i B ) = ( B i^i ( Base ` W ) ) )
4	2, 3	eqtr3id 2243	. . . . . . 7 |- ( ph -> B = ( B i^i ( Base ` W ) ) )
5	1, 4	eqtr3d 2231	. . . . . 6 |- ( ph -> ( Base ` W ) = ( B i^i ( Base ` W ) ) )
6	5	ineq2d 3365	. . . . 5 |- ( ph -> ( A i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) ) )
7		inass 3374	. . . . 5 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
8	6, 7	eqtr4di 2247	. . . 4 |- ( ph -> ( A i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) ) )
9	8	opeq2d 3816	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. )
10	9	oveq2d 5941	. 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 12767	. . 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 4170	. . . 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 12767	. . 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 2239	1 |- ( ph -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   i^i cin 3156   <.cop 3626   ` cfv 5259  (class class class)co 5925   ndxcnx 12700   sSet csts 12701   Basecbs 12703   ↾_s cress 12704

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711

This theorem is referenced by: (None)