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Theorem strss 17145
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 𝑆 βŠ† 𝑇
strss.e 𝐸 = Slot (πΈβ€˜ndx)
strss.n ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆
Assertion
Ref Expression
strss (πΈβ€˜π‘‡) = (πΈβ€˜π‘†)

Proof of Theorem strss
StepHypRef Expression
1 strss.e . . 3 𝐸 = Slot (πΈβ€˜ndx)
2 strss.t . . . 4 𝑇 ∈ V
32a1i 11 . . 3 (⊀ β†’ 𝑇 ∈ V)
4 strss.f . . . 4 Fun 𝑇
54a1i 11 . . 3 (⊀ β†’ Fun 𝑇)
6 strss.s . . . 4 𝑆 βŠ† 𝑇
76a1i 11 . . 3 (⊀ β†’ 𝑆 βŠ† 𝑇)
8 strss.n . . . 4 ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆
98a1i 11 . . 3 (⊀ β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)
101, 3, 5, 7, 9strssd 17144 . 2 (⊀ β†’ (πΈβ€˜π‘‡) = (πΈβ€˜π‘†))
1110mptru 1540 1 (πΈβ€˜π‘‡) = (πΈβ€˜π‘†)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  βŠ€wtru 1534   ∈ wcel 2098  Vcvv 3466   βŠ† wss 3941  βŸ¨cop 4627  Fun wfun 6528  β€˜cfv 6534  Slot cslot 17119  ndxcnx 17131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-slot 17120
This theorem is referenced by:  grpss  18880
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