Theorem ressval2 12019

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 982	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2699	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 983	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2699	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 981	. . . . . 6 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> -. B C_ A )
7	6	iffalsed 3484	. . . . 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 12014	. . . . . . 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 4064	. . . . . . 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 11991	. . . . . 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 1216	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) e. _V )
14	7, 13	eqeltrd 2216	. . . 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 5425	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
17		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
18	16, 17	syl6eqr 2190	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
19		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
20	18, 19	sseq12d 3128	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
21	19, 18	ineq12d 3278	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
22	21	opeq2d 3712	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
23	15, 22	oveq12d 5792	. . . . . 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 3498	. . . . 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 11967	. . . . 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 5900	. . . 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 1216	. . 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 2172	. 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 2184	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 962   = wceq 1331   e. wcel 1480   _Vcvv 2686   i^i cin 3070   C_ wss 3071   ifcif 3474   <.cop 3530   ` cfv 5123  (class class class)co 5774   NNcn 8720   ndxcnx 11956   sSet csts 11957   Basecbs 11959   ↾_s cress 11960

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-inn 8721  df-ndx 11962  df-slot 11963  df-base 11965  df-sets 11966  df-ress 11967

This theorem is referenced by: (None)