Theorem ressinbasd 13371

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 3434	. . . . . . . 8 |- ( B i^i B ) = B
3	1	ineq2d 3426	. . . . . . . 8 |- ( ph -> ( B i^i B ) = ( B i^i ( Base ` W ) ) )
4	2, 3	eqtr3id 2281	. . . . . . 7 |- ( ph -> B = ( B i^i ( Base ` W ) ) )
5	1, 4	eqtr3d 2269	. . . . . 6 |- ( ph -> ( Base ` W ) = ( B i^i ( Base ` W ) ) )
6	5	ineq2d 3426	. . . . 5 |- ( ph -> ( A i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) ) )
7		inass 3435	. . . . 5 |- ( ( A i^i B ) i^i ( Base ` W ) ) = ( A i^i ( B i^i ( Base ` W ) ) )
8	6, 7	eqtr4di 2285	. . . 4 |- ( ph -> ( A i^i ( Base ` W ) ) = ( ( A i^i B ) i^i ( Base ` W ) ) )
9	8	opeq2d 3895	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( ( A i^i B ) i^i ( Base ` W ) ) >. )
10	9	oveq2d 6074	. 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 13361	. . 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 4251	. . . 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 13361	. . 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 2277	1 |- ( ph -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   i^i cin 3213   <.cop 3697   ` cfv 5357  (class class class)co 6058   ndxcnx 13293   sSet csts 13294   Basecbs 13296   ↾_s cress 13297

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

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 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304

This theorem is referenced by: (None)