Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-taginv Structured version   Visualization version   GIF version

Theorem bj-taginv 33849
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 33835 . 2 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2 bj-sngltag 33846 . . . 4 (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
32elv 3415 . . 3 ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
43abbii 2839 . 2 {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
51, 4eqtri 2797 1 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1508  wcel 2051  {cab 2753  Vcvv 3410  {csn 4436  sngl bj-csngl 33828  tag bj-ctag 33837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-rex 3089  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-sn 4437  df-pr 4439  df-bj-sngl 33829  df-bj-tag 33838
This theorem is referenced by:  bj-projval  33859
  Copyright terms: Public domain W3C validator