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 Description: Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
Hypotheses
Ref Expression
Assertion
Ref Expression

StepHypRef Expression
1 strss.e . . 3 𝐸 = Slot (𝐸‘ndx)
2 strss.t . . . 4 𝑇 ∈ V
32a1i 11 . . 3 (⊤ → 𝑇 ∈ V)
4 strss.f . . . 4 Fun 𝑇
54a1i 11 . . 3 (⊤ → Fun 𝑇)
6 strss.s . . . 4 𝑆𝑇
76a1i 11 . . 3 (⊤ → 𝑆𝑇)
8 strss.n . . . 4 ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
98a1i 11 . . 3 (⊤ → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)
101, 3, 5, 7, 9strssd 15737 . 2 (⊤ → (𝐸𝑇) = (𝐸𝑆))
1110trud 1484 1 (𝐸𝑇) = (𝐸𝑆)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131  Fun wfun 5798  ‘cfv 5804  ndxcnx 15692  Slot cslot 15694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-slot 15699 This theorem is referenced by:  grpss  17263
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