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Theorem setsstrset 36012
Description: Relation between df-sets 17096 and df-strset 36011. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.)
Assertion
Ref Expression
setsstrset ((𝑆 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ [𝐡 / 𝐴]struct𝑆 = (𝑆 sSet ⟨(π΄β€˜ndx), 𝐡⟩))

Proof of Theorem setsstrset
StepHypRef Expression
1 df-strset 36011 . 2 [𝐡 / 𝐴]struct𝑆 = ((𝑆 β†Ύ (V βˆ– {(π΄β€˜ndx)})) βˆͺ {⟨(π΄β€˜ndx), 𝐡⟩})
2 setsval 17099 . 2 ((𝑆 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝑆 sSet ⟨(π΄β€˜ndx), 𝐡⟩) = ((𝑆 β†Ύ (V βˆ– {(π΄β€˜ndx)})) βˆͺ {⟨(π΄β€˜ndx), 𝐡⟩}))
31, 2eqtr4id 2791 1 ((𝑆 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ [𝐡 / 𝐴]struct𝑆 = (𝑆 sSet ⟨(π΄β€˜ndx), 𝐡⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βˆ– cdif 3945   βˆͺ cun 3946  {csn 4628  βŸ¨cop 4634   β†Ύ cres 5678  β€˜cfv 6543  (class class class)co 7408   sSet csts 17095  ndxcnx 17125  [cstrset 36010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-sets 17096  df-strset 36011
This theorem is referenced by: (None)
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