Theorem ressval2 12058

Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Hypotheses

Ref	Expression
ressbas.r	|- R = ( W ↾s A )
ressbas.b	|- B = ( Base ` W )

Assertion

Ref	Expression
ressval2	|- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )

Proof of Theorem ressval2

Dummy variables w a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressbas.r	. 2 |- R = ( W ↾s A )
2		simp2 983	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2702	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 984	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2702	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 982	. . . . . 6 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> -. B C_ A )
7	6	iffalsed 3489	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
8		basendxnn 12053	. . . . . . 7 |- ( Base ` ndx ) e. NN
9	8	a1i 9	. . . . . 6 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( Base ` ndx ) e. NN )
10		inex1g 4072	. . . . . . 7 |- ( A e. Y -> ( A i^i B ) e. _V )
11	4, 10	syl 14	. . . . . 6 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( A i^i B ) e. _V )
12		setsex 12030	. . . . . 6 |- ( ( W e. X /\ ( Base ` ndx ) e. NN /\ ( A i^i B ) e. _V ) -> ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) e. _V )
13	2, 9, 11, 12	syl3anc 1217	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) e. _V )
14	7, 13	eqeltrd 2217	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) e. _V )
15		simpl 108	. . . . . . . . 9 |- ( ( w = W /\ a = A ) -> w = W )
16	15	fveq2d 5433	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
17		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
18	16, 17	eqtr4di 2191	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
19		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
20	18, 19	sseq12d 3133	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
21	19, 18	ineq12d 3283	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
22	21	opeq2d 3720	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
23	15, 22	oveq12d 5800	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( w sSet <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
24	20, 15, 23	ifbieq12d 3503	. . . . 5 |- ( ( w = W /\ a = A ) -> if ( ( Base ` w ) C_ a , w , ( w sSet <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. ) ) = if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) )
25		df-ress 12006	. . . . 5 |- ↾s = ( w e. _V , a e. _V |-> if ( ( Base ` w ) C_ a , w , ( w sSet <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. ) ) )
26	24, 25	ovmpoga 5908	. . . 4 |- ( ( W e. _V /\ A e. _V /\ if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) e. _V ) -> ( W ↾s A ) = if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) )
27	3, 5, 14, 26	syl3anc 1217	. . 3 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = if ( B C_ A , W , ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) ) )
28	27, 7	eqtrd 2173	. 2 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
29	1, 28	syl5eq 2185	1 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481   _Vcvv 2689   i^i cin 3075   C_ wss 3076   ifcif 3479   <.cop 3535   ` cfv 5131  (class class class)co 5782   NNcn 8744   ndxcnx 11995   sSet csts 11996   Basecbs 11998   ↾_s cress 11999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1re 7738  ax-addrcl 7741

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-inn 8745  df-ndx 12001  df-slot 12002  df-base 12004  df-sets 12005  df-ress 12006

This theorem is referenced by: (None)