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Theorem setsstrset 36671
Description: Relation between df-sets 17130 and df-strset 36670. 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 36670 . 2 [𝐡 / 𝐴]struct𝑆 = ((𝑆 β†Ύ (V βˆ– {(π΄β€˜ndx)})) βˆͺ {⟨(π΄β€˜ndx), 𝐡⟩})
2 setsval 17133 . 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 2784 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   βˆ– cdif 3937   βˆͺ cun 3938  {csn 4624  βŸ¨cop 4630   β†Ύ cres 5674  β€˜cfv 6542  (class class class)co 7415   sSet csts 17129  ndxcnx 17159  [cstrset 36669
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-sets 17130  df-strset 36670
This theorem is referenced by: (None)
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