Theorem ressval2 12455

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 988	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2739	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 989	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2739	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 987	. . . . . 6 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> -. B C_ A )
7	6	iffalsed 3530	. . . . 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 12449	. . . . . . 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 4118	. . . . . . 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 12426	. . . . . 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 1228	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) e. _V )
14	7, 13	eqeltrd 2243	. . . 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 5490	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
17		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
18	16, 17	eqtr4di 2217	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
19		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
20	18, 19	sseq12d 3173	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
21	19, 18	ineq12d 3324	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
22	21	opeq2d 3765	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
23	15, 22	oveq12d 5860	. . . . . 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 3546	. . . . 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 12402	. . . . 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 5971	. . . 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 1228	. . 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 2198	. 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 2211	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 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116   ifcif 3520   <.cop 3579   ` cfv 5188  (class class class)co 5842   NNcn 8857   ndxcnx 12391   sSet csts 12392   Basecbs 12394   ↾_s cress 12395

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-inn 8858  df-ndx 12397  df-slot 12398  df-base 12400  df-sets 12401  df-ress 12402

This theorem is referenced by: (None)