Theorem ressval2 12478

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 993	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. X )
3	2	elexd 2743	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> W e. _V )
4		simp3 994	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. Y )
5	4	elexd 2743	. . . 4 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> A e. _V )
6		simp1 992	. . . . . 6 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> -. B C_ A )
7	6	iffalsed 3536	. . . . 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 12471	. . . . . . 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 4125	. . . . . . 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 12448	. . . . . 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 1233	. . . . 5 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) e. _V )
14	7, 13	eqeltrd 2247	. . . 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 5500	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = ( Base ` W ) )
17		ressbas.b	. . . . . . . 8 |- B = ( Base ` W )
18	16, 17	eqtr4di 2221	. . . . . . 7 |- ( ( w = W /\ a = A ) -> ( Base ` w ) = B )
19		simpr 109	. . . . . . 7 |- ( ( w = W /\ a = A ) -> a = A )
20	18, 19	sseq12d 3178	. . . . . 6 |- ( ( w = W /\ a = A ) -> ( ( Base ` w ) C_ a <-> B C_ A ) )
21	19, 18	ineq12d 3329	. . . . . . . 8 |- ( ( w = W /\ a = A ) -> ( a i^i ( Base ` w ) ) = ( A i^i B ) )
22	21	opeq2d 3772	. . . . . . 7 |- ( ( w = W /\ a = A ) -> <. ( Base ` ndx ) , ( a i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( A i^i B ) >. )
23	15, 22	oveq12d 5871	. . . . . 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 3552	. . . . 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 12424	. . . . 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 5982	. . . 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 1233	. . 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 2203	. 2 |- ( ( -. B C_ A /\ W e. X /\ A e. Y ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i B ) >. ) )
29	1, 28	eqtrid 2215	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 973   = wceq 1348   e. wcel 2141   _Vcvv 2730   i^i cin 3120   C_ wss 3121   ifcif 3526   <.cop 3586   ` cfv 5198  (class class class)co 5853   NNcn 8878   ndxcnx 12413   sSet csts 12414   Basecbs 12416   ↾_s cress 12417

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-sets 12423  df-ress 12424

This theorem is referenced by: (None)