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Theorem resvval 27970
 Description: Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Hypotheses
Ref Expression
resvsca.r v
resvsca.f Scalar
resvsca.b
Assertion
Ref Expression
resvval sSet Scalar s

Proof of Theorem resvval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resvsca.r . 2 v
2 elex 3118 . . 3
3 elex 3118 . . 3
4 ovex 6324 . . . . . 6 sSet Scalar s
5 ifcl 3986 . . . . . 6 sSet Scalar s sSet Scalar s
64, 5mpan2 671 . . . . 5 sSet Scalar s
76adantr 465 . . . 4 sSet Scalar s
8 simpl 457 . . . . . . . . . . 11
98fveq2d 5876 . . . . . . . . . 10 Scalar Scalar
10 resvsca.f . . . . . . . . . 10 Scalar
119, 10syl6eqr 2516 . . . . . . . . 9 Scalar
1211fveq2d 5876 . . . . . . . 8 Scalar
13 resvsca.b . . . . . . . 8
1412, 13syl6eqr 2516 . . . . . . 7 Scalar
15 simpr 461 . . . . . . 7
1614, 15sseq12d 3528 . . . . . 6 Scalar
1711, 15oveq12d 6314 . . . . . . . 8 Scalars s
1817opeq2d 4226 . . . . . . 7 Scalar Scalars Scalar s
198, 18oveq12d 6314 . . . . . 6 sSet Scalar Scalars sSet Scalar s
2016, 8, 19ifbieq12d 3971 . . . . 5 Scalar sSet Scalar Scalars sSet Scalar s
21 df-resv 27968 . . . . 5 v Scalar sSet Scalar Scalars
2220, 21ovmpt2ga 6431 . . . 4 sSet Scalar s v sSet Scalar s
237, 22mpd3an3 1325 . . 3 v sSet Scalar s
242, 3, 23syl2an 477 . 2 v sSet Scalar s
251, 24syl5eq 2510 1 sSet Scalar s
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819  cvv 3109   wss 3471  cif 3944  cop 4038  cfv 5594  (class class class)co 6296  cnx 14640   sSet csts 14641  cbs 14643   ↾s cress 14644  Scalarcsca 14714   ↾v cresv 27967 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-resv 27968 This theorem is referenced by:  resvid2  27971  resvval2  27972
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