Theorem strressid 13120

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 3864	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( B i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( Base ` W ) >. )
6	5	oveq2d 6023	. 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 13060	. . . 4 |- ( W Struct <. M , N >. -> W e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> W e. _V )
10		basfn 13107	. . . . 5 |- Base Fn _V
11		funfvex 5646	. . . . . 6 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
12	11	funfni 5423	. . . . 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 13113	. . 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 13102	. . 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 13081	. 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 4083   dom cdm 4719   Fun wfun 5312   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   Struct cstr 13044   ndxcnx 13045   sSet csts 13046   Basecbs 13048   ↾_s cress 13049

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 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126

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 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-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-struct 13050  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-iress 13056

This theorem is referenced by: (None)