Theorem strressid 13099

Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)

Hypotheses

Ref	Expression
strressid.b	|- ( ph -> B = ( Base ` W ) )
strressid.s	|- ( ph -> W Struct <. M , N >. )
strressid.f	|- ( ph -> Fun W )
strressid.bw	|- ( ph -> ( Base ` ndx ) e. dom W )

Assertion

Ref	Expression
strressid	|- ( ph -> ( W ↾s B ) = W )

Proof of Theorem strressid

Step	Hyp	Ref	Expression
1		strressid.b	. . . . . 6 |- ( ph -> B = ( Base ` W ) )
2	1	ineq1d 3404	. . . . 5 |- ( ph -> ( B i^i ( Base ` W ) ) = ( ( Base ` W ) i^i ( Base ` W ) ) )
3		inidm 3413	. . . . 5 |- ( ( Base ` W ) i^i ( Base ` W ) ) = ( Base ` W )
4	2, 3	eqtrdi 2278	. . . 4 |- ( ph -> ( B i^i ( Base ` W ) ) = ( Base ` W ) )
5	4	opeq2d 3863	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( B i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( Base ` W ) >. )
6	5	oveq2d 6016	. 2 |- ( ph -> ( W sSet <. ( Base ` ndx ) , ( B i^i ( Base ` W ) ) >. ) = ( W sSet <. ( Base ` ndx ) , ( Base ` W ) >. ) )
7		strressid.s	. . . 4 |- ( ph -> W Struct <. M , N >. )
8		structex 13039	. . . 4 |- ( W Struct <. M , N >. -> W e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> W e. _V )
10		basfn 13086	. . . . 5 |- Base Fn _V
11		funfvex 5643	. . . . . 6 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
12	11	funfni 5422	. . . . 5 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
13	10, 9, 12	sylancr 414	. . . 4 |- ( ph -> ( Base ` W ) e. _V )
14	1, 13	eqeltrd 2306	. . 3 |- ( ph -> B e. _V )
15		ressvalsets 13092	. . 3 |- ( ( W e. _V /\ B e. _V ) -> ( W ↾s B ) = ( W sSet <. ( Base ` ndx ) , ( B i^i ( Base ` W ) ) >. ) )
16	9, 14, 15	syl2anc 411	. 2 |- ( ph -> ( W ↾s B ) = ( W sSet <. ( Base ` ndx ) , ( B i^i ( Base ` W ) ) >. ) )
17		baseid 13081	. . 3 |- Base = Slot ( Base ` ndx )
18		strressid.f	. . 3 |- ( ph -> Fun W )
19		strressid.bw	. . 3 |- ( ph -> ( Base ` ndx ) e. dom W )
20	17, 7, 18, 19	strsetsid 13060	. 2 |- ( ph -> W = ( W sSet <. ( Base ` ndx ) , ( Base ` W ) >. ) )
21	6, 16, 20	3eqtr4d 2272	1 |- ( ph -> ( W ↾s B ) = W )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   <.cop 3669   class class class wbr 4082   dom cdm 4718   Fun wfun 5311   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   Struct cstr 13023   ndxcnx 13024   sSet csts 13025   Basecbs 13027   ↾_s cress 13028

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  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-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  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 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-struct 13029  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035

This theorem is referenced by: (None)