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Theorem elcnvcnvintab 43537
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 6209 . . . 4 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 4217 . . . 4 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2761 . . 3 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
43eleq2i 2829 . 2 (𝐴 {𝑥𝜑} ↔ 𝐴 ∈ ((V × V) ∩ {𝑥𝜑}))
5 elinintab 43530 . 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 206  wa 395  wal 1533  wcel 2104  {cab 2710  Vcvv 3477  cin 3962   cint 4954   × cxp 5682  ccnv 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-int 4955  df-br 5151  df-opab 5213  df-xp 5690  df-rel 5691  df-cnv 5692
This theorem is referenced by:  cnvcnvintabd  43555
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