Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elcnvcnvintab Structured version   Visualization version   GIF version

Theorem elcnvcnvintab 42319
Description: Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
Assertion
Ref Expression
elcnvcnvintab (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elcnvcnvintab
StepHypRef Expression
1 cnvcnv 6189 . . . 4 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 4201 . . . 4 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2761 . . 3 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
43eleq2i 2826 . 2 (𝐴 {𝑥𝜑} ↔ 𝐴 ∈ ((V × V) ∩ {𝑥𝜑}))
5 elinintab 42312 . 2 (𝐴 ∈ ((V × V) ∩ {𝑥𝜑}) ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
64, 5bitri 275 1 (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wcel 2107  {cab 2710  Vcvv 3475  cin 3947   cint 4950   × cxp 5674  ccnv 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  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-int 4951  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684
This theorem is referenced by:  cnvcnvintabd  42337
  Copyright terms: Public domain W3C validator