Theorem strressid 13284

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 3421	. . . . 5 |- ( ph -> ( B i^i ( Base ` W ) ) = ( ( Base ` W ) i^i ( Base ` W ) ) )
3		inidm 3430	. . . . 5 |- ( ( Base ` W ) i^i ( Base ` W ) ) = ( Base ` W )
4	2, 3	eqtrdi 2281	. . . 4 |- ( ph -> ( B i^i ( Base ` W ) ) = ( Base ` W ) )
5	4	opeq2d 3890	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( B i^i ( Base ` W ) ) >. = <. ( Base ` ndx ) , ( Base ` W ) >. )
6	5	oveq2d 6066	. 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 13224	. . . 4 |- ( W Struct <. M , N >. -> W e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> W e. _V )
10		basfn 13271	. . . . 5 |- Base Fn _V
11		funfvex 5687	. . . . . 6 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
12	11	funfni 5458	. . . . 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 2309	. . 3 |- ( ph -> B e. _V )
15		ressvalsets 13277	. . 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 13266	. . 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 13245	. 2 |- ( ph -> W = ( W sSet <. ( Base ` ndx ) , ( Base ` W ) >. ) )
21	6, 16, 20	3eqtr4d 2275	1 |- ( ph -> ( W ↾s B ) = W )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   <.cop 3692   class class class wbr 4109   dom cdm 4749   Fun wfun 5346   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Struct cstr 13208   ndxcnx 13209   sSet csts 13210   Basecbs 13212   ↾_s cress 13213

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-struct 13214  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220

This theorem is referenced by: (None)