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Theorem bj-taginv 34195
Description: Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-taginv 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-taginv
StepHypRef Expression
1 bj-snglinv 34181 . 2 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2 bj-sngltag 34192 . . . 4 (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
32elv 3497 . . 3 ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
43abbii 2883 . 2 {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
51, 4eqtri 2841 1 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492  {csn 4557  sngl bj-csngl 34174  tag bj-ctag 34183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-bj-sngl 34175  df-bj-tag 34184
This theorem is referenced by:  bj-projval  34205
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