Theorem ressvalsets 12767

Description: Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)

Assertion

Ref	Expression
ressvalsets	|- ( ( W e. X /\ A e. Y ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )

Proof of Theorem ressvalsets

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

Step	Hyp	Ref	Expression
1		elex 2774	. . 3 |- ( W e. X -> W e. _V )
2	1	adantr 276	. 2 |- ( ( W e. X /\ A e. Y ) -> W e. _V )
3		elex 2774	. . 3 |- ( A e. Y -> A e. _V )
4	3	adantl 277	. 2 |- ( ( W e. X /\ A e. Y ) -> A e. _V )
5		simpl 109	. . 3 |- ( ( W e. X /\ A e. Y ) -> W e. X )
6		basendxnn 12759	. . . 4 |- ( Base ` ndx ) e. NN
7	6	a1i 9	. . 3 |- ( ( W e. X /\ A e. Y ) -> ( Base ` ndx ) e. NN )
8		inex1g 4170	. . . 4 |- ( A e. Y -> ( A i^i ( Base ` W ) ) e. _V )
9	8	adantl 277	. . 3 |- ( ( W e. X /\ A e. Y ) -> ( A i^i ( Base ` W ) ) e. _V )
10		setsex 12735	. . 3 |- ( ( W e. X /\ ( Base ` ndx ) e. NN /\ ( A i^i ( Base ` W ) ) e. _V ) -> ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) e. _V )
11	5, 7, 9, 10	syl3anc 1249	. 2 |- ( ( W e. X /\ A e. Y ) -> ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) e. _V )
12		id 19	. . . 4 |- ( w = W -> w = W )
13		fveq2 5561	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
14	13	ineq2d 3365	. . . . 5 |- ( w = W -> ( x i^i ( Base ` w ) ) = ( x i^i ( Base ` W ) ) )
15	14	opeq2d 3816	. . . 4 |- ( w = W -> <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. )
16	12, 15	oveq12d 5943	. . 3 |- ( w = W -> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. ) )
17		ineq1 3358	. . . . 5 |- ( x = A -> ( x i^i ( Base ` W ) ) = ( A i^i ( Base ` W ) ) )
18	17	opeq2d 3816	. . . 4 |- ( x = A -> <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. )
19	18	oveq2d 5941	. . 3 |- ( x = A -> ( W sSet <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
20		df-iress 12711	. . 3 |- ↾s = ( w e. _V , x e. _V |-> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
21	16, 19, 20	ovmpog 6061	. 2 |- ( ( W e. _V /\ A e. _V /\ ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) e. _V ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
22	2, 4, 11, 21	syl3anc 1249	1 |- ( ( W e. X /\ A e. Y ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   i^i cin 3156   <.cop 3626   ` cfv 5259  (class class class)co 5925   NNcn 9007   ndxcnx 12700   sSet csts 12701   Basecbs 12703   ↾_s cress 12704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711

This theorem is referenced by:  ressex  12768  ressval2  12769  ressbasd  12770  strressid  12774  ressval3d  12775  resseqnbasd  12776  ressinbasd  12777  ressressg  12778  mgpress  13563