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Theorem elcnvcnvintab 36707
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 5488 . . . 4 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 3763 . . . 4 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2628 . . 3 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
43eleq2i 2676 . 2 (𝐴 {𝑥𝜑} ↔ 𝐴 ∈ ((V × V) ∩ {𝑥𝜑}))
5 elinintab 36700 . 2 (𝐴 ∈ ((V × V) ∩ {𝑥𝜑}) ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
64, 5bitri 262 1 (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wcel 1976  {cab 2592  Vcvv 3169  cin 3535   cint 4401   × cxp 5023  ccnv 5024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-int 4402  df-br 4575  df-opab 4635  df-xp 5031  df-rel 5032  df-cnv 5033
This theorem is referenced by:  cnvcnvintabd  36725
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