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Theorem bj-taginv 32642
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 32628 . 2 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2 vex 3189 . . . 4 𝑥 ∈ V
3 bj-sngltag 32639 . . . 4 (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
42, 3ax-mp 5 . . 3 ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
54abbii 2736 . 2 {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
61, 5eqtri 2643 1 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  {csn 4150  sngl bj-csngl 32621  tag bj-ctag 32630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-sn 4151  df-pr 4153  df-bj-sngl 32622  df-bj-tag 32631
This theorem is referenced by:  bj-projval  32652
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