Theorem ressvalsets 13097

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 2811	. . 3 |- ( W e. X -> W e. _V )
2	1	adantr 276	. 2 |- ( ( W e. X /\ A e. Y ) -> W e. _V )
3		elex 2811	. . 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 13088	. . . 4 |- ( Base ` ndx ) e. NN
7	6	a1i 9	. . 3 |- ( ( W e. X /\ A e. Y ) -> ( Base ` ndx ) e. NN )
8		inex1g 4220	. . . 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 13064	. . 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 1271	. 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 5627	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
14	13	ineq2d 3405	. . . . 5 |- ( w = W -> ( x i^i ( Base ` w ) ) = ( x i^i ( Base ` W ) ) )
15	14	opeq2d 3864	. . . 4 |- ( w = W -> <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. = <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. )
16	12, 15	oveq12d 6019	. . 3 |- ( w = W -> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. ) )
17		ineq1 3398	. . . . 5 |- ( x = A -> ( x i^i ( Base ` W ) ) = ( A i^i ( Base ` W ) ) )
18	17	opeq2d 3864	. . . 4 |- ( x = A -> <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. )
19	18	oveq2d 6017	. . 3 |- ( x = A -> ( W sSet <. ( Base ` ndx ) , ( x i^i ( Base ` W ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
20		df-iress 13040	. . 3 |- ↾s = ( w e. _V , x e. _V |-> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
21	16, 19, 20	ovmpog 6139	. 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 1271	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 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   <.cop 3669   ` cfv 5318  (class class class)co 6001   NNcn 9110   ndxcnx 13029   sSet csts 13030   Basecbs 13032   ↾_s cress 13033

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-inn 9111  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040

This theorem is referenced by:  ressex  13098  ressval2  13099  ressbasd  13100  strressid  13104  ressval3d  13105  resseqnbasd  13106  ressinbasd  13107  ressressg  13108  mgpress  13894