Theorem ressinbas 15936

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.)

Hypothesis

Ref	Expression
ressid.1	|- B = ( Base ` W )

Assertion

Ref	Expression
ressinbas	|- ( A e. X -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )

Proof of Theorem ressinbas

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. X -> A e. _V )
2		eqid 2622	. . . . . . 7 |- ( W ↾s A ) = ( W ↾s A )
3		ressid.1	. . . . . . 7 |- B = ( Base ` W )
4	2, 3	ressid2 15928	. . . . . 6 |- ( ( B C_ A /\ W e. _V /\ A e. _V ) -> ( W ↾s A ) = W )
5		ssid 3624	. . . . . . . 8 |- B C_ B
6		incom 3805	. . . . . . . . 9 |- ( A i^i B ) = ( B i^i A )
7		df-ss 3588	. . . . . . . . . 10 |- ( B C_ A <-> ( B i^i A ) = B )
8	7	biimpi 206	. . . . . . . . 9 |- ( B C_ A -> ( B i^i A ) = B )
9	6, 8	syl5eq 2668	. . . . . . . 8 |- ( B C_ A -> ( A i^i B ) = B )
10	5, 9	syl5sseqr 3654	. . . . . . 7 |- ( B C_ A -> B C_ ( A i^i B ) )
11		elex 3212	. . . . . . 7 |- ( W e. _V -> W e. _V )
12		inex1g 4801	. . . . . . 7 |- ( A e. _V -> ( A i^i B ) e. _V )
13		eqid 2622	. . . . . . . 8 |- ( W ↾s ( A i^i B ) ) = ( W ↾s ( A i^i B ) )
14	13, 3	ressid2 15928	. . . . . . 7 |- ( ( B C_ ( A i^i B ) /\ W e. _V /\ ( A i^i B ) e. _V ) -> ( W ↾s ( A i^i B ) ) = W )
15	10, 11, 12, 14	syl3an 1368	. . . . . 6 |- ( ( B C_ A /\ W e. _V /\ A e. _V ) -> ( W ↾s ( A i^i B ) ) = W )
16	4, 15	eqtr4d 2659	. . . . 5 |- ( ( B C_ A /\ W e. _V /\ A e. _V ) -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
17	16	3expb 1266	. . . 4 |- ( ( B C_ A /\ ( W e. _V /\ A e. _V ) ) -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
18		inass 3823	. . . . . . . . 9 |- ( ( A i^i B ) i^i B ) = ( A i^i ( B i^i B ) )
19		inidm 3822	. . . . . . . . . 10 |- ( B i^i B ) = B
20	19	ineq2i 3811	. . . . . . . . 9 |- ( A i^i ( B i^i B ) ) = ( A i^i B )
21	18, 20	eqtr2i 2645	. . . . . . . 8 |- ( A i^i B ) = ( ( A i^i B ) i^i B )
22	21	opeq2i 4406	. . . . . . 7 |- <. ( Base ` ndx ) , ( A i^i B ) >. = <. ( Base ` ndx ) , ( ( A i^i B ) i^i B ) >.
23	22	oveq2i 6661	. . . . . 6 |- ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i B ) >. )
24	2, 3	ressval2 15929	. . . . . 6 |- ( ( -. B C_ A /\ W e. _V /\ A e. _V ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
25		inss1 3833	. . . . . . . . 9 |- ( A i^i B ) C_ A
26		sstr 3611	. . . . . . . . 9 |- ( ( B C_ ( A i^i B ) /\ ( A i^i B ) C_ A ) -> B C_ A )
27	25, 26	mpan2 707	. . . . . . . 8 |- ( B C_ ( A i^i B ) -> B C_ A )
28	27	con3i 150	. . . . . . 7 |- ( -. B C_ A -> -. B C_ ( A i^i B ) )
29	13, 3	ressval2 15929	. . . . . . 7 |- ( ( -. B C_ ( A i^i B ) /\ 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 B ) >. ) )
30	28, 11, 12, 29	syl3an 1368	. . . . . 6 |- ( ( -. B C_ A /\ W e. _V /\ A e. _V ) -> ( W ↾s ( A i^i B ) ) = ( W sSet <. ( Base ` ndx ) , ( ( A i^i B ) i^i B ) >. ) )
31	23, 24, 30	3eqtr4a 2682	. . . . 5 |- ( ( -. B C_ A /\ W e. _V /\ A e. _V ) -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
32	31	3expb 1266	. . . 4 |- ( ( -. B C_ A /\ ( W e. _V /\ A e. _V ) ) -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
33	17, 32	pm2.61ian 831	. . 3 |- ( ( W e. _V /\ A e. _V ) -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
34		reldmress 15926	. . . . . 6 |- Rel dom ↾s
35	34	ovprc1 6684	. . . . 5 |- ( -. W e. _V -> ( W ↾s A ) = (/) )
36	34	ovprc1 6684	. . . . 5 |- ( -. W e. _V -> ( W ↾s ( A i^i B ) ) = (/) )
37	35, 36	eqtr4d 2659	. . . 4 |- ( -. W e. _V -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
38	37	adantr 481	. . 3 |- ( ( -. W e. _V /\ A e. _V ) -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
39	33, 38	pm2.61ian 831	. 2 |- ( A e. _V -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )
40	1, 39	syl 17	1 |- ( A e. X -> ( W ↾s A ) = ( W ↾s ( A i^i B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ress 15865

This theorem is referenced by:  ressress  15938  rescabs  16493  resscat  16512  funcres2c  16561  ressffth  16598  cphsubrglem  22977  suborng  29815