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Theorem strss 16232
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 16231 . 2 (⊤ → (𝐸𝑇) = (𝐸𝑆))
1110mptru 1661 1 (𝐸𝑇) = (𝐸𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wtru 1654  wcel 2157  Vcvv 3384  wss 3768  cop 4373  Fun wfun 6094  cfv 6100  ndxcnx 16178  Slot cslot 16180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6063  df-fun 6102  df-fv 6108  df-slot 16185
This theorem is referenced by:  grpss  17753
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