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Theorem strss 15957
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
strss.t 𝑇 ∈ V
strss.f Fun 𝑇
strss.s 𝑆𝑇
strss.e 𝐸 = Slot (𝐸‘ndx)
strss.n ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆
Assertion
Ref Expression
strss (𝐸𝑇) = (𝐸𝑆)

Proof of Theorem strss
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 15956 . 2 (⊤ → (𝐸𝑇) = (𝐸𝑆))
1110trud 1533 1 (𝐸𝑇) = (𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wtru 1524  wcel 2030  Vcvv 3231  wss 3607  cop 4216  Fun wfun 5920  cfv 5926  ndxcnx 15901  Slot cslot 15903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-slot 15908
This theorem is referenced by:  grpss  17487
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