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Theorem elcnvcnvlem 42336
Description: Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.)
Assertion
Ref Expression
elcnvcnvlem (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))

Proof of Theorem elcnvcnvlem
StepHypRef Expression
1 cnvcnv 6189 . . . 4 𝐵 = (𝐵 ∩ (V × V))
2 incom 4201 . . . 4 (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵)
31, 2eqtri 2761 . . 3 𝐵 = ((V × V) ∩ 𝐵)
43eleq2i 2826 . 2 (𝐴𝐵𝐴 ∈ ((V × V) ∩ 𝐵))
5 elinlem 42335 . 2 (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
64, 5bitri 275 1 (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  Vcvv 3475  cin 3947   I cid 5573   × cxp 5674  ccnv 5675  cfv 6541
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-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-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  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-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6493  df-fun 6543  df-fv 6549
This theorem is referenced by: (None)
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