Theorem ressvalsets 13011

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 2788	. . 3 |- ( W e. X -> W e. _V )
2	1	adantr 276	. 2 |- ( ( W e. X /\ A e. Y ) -> W e. _V )
3		elex 2788	. . 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 13003	. . . 4 |- ( Base ` ndx ) e. NN
7	6	a1i 9	. . 3 |- ( ( W e. X /\ A e. Y ) -> ( Base ` ndx ) e. NN )
8		inex1g 4196	. . . 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 12979	. . 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 1250	. 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 5599	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
14	13	ineq2d 3382	. . . . 5 |- ( w = W -> ( x i^i ( Base ` w ) ) = ( x i^i ( Base ` W ) ) )
15	14	opeq2d 3840	. . . 4 |- ( w = W -> <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. )
16	12, 15	oveq12d 5985	. . 3 |- ( w = W -> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. ) )
17		ineq1 3375	. . . . 5 |- ( x = A -> ( x i^i ( Base ` W ) ) = ( A i^i ( Base ` W ) ) )
18	17	opeq2d 3840	. . . 4 |- ( x = A -> <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. )
19	18	oveq2d 5983	. . 3 |- ( x = A -> ( W sSet <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
20		df-iress 12955	. . 3 |- ↾s = ( w e. _V , x e. _V |-> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
21	16, 19, 20	ovmpog 6103	. 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 1250	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 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   <.cop 3646   ` cfv 5290  (class class class)co 5967   NNcn 9071   ndxcnx 12944   sSet csts 12945   Basecbs 12947   ↾_s cress 12948

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955

This theorem is referenced by:  ressex  13012  ressval2  13013  ressbasd  13014  strressid  13018  ressval3d  13019  resseqnbasd  13020  ressinbasd  13021  ressressg  13022  mgpress  13808