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Theorem elcnvcnvlem 39327
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 5889 . . . 4 𝐵 = (𝐵 ∩ (V × V))
2 incom 4066 . . . 4 (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵)
31, 2eqtri 2802 . . 3 𝐵 = ((V × V) ∩ 𝐵)
43eleq2i 2857 . 2 (𝐴𝐵𝐴 ∈ ((V × V) ∩ 𝐵))
5 elinlem 39326 . 2 (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
64, 5bitri 267 1 (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wcel 2050  Vcvv 3415  cin 3828   I cid 5311   × cxp 5405  ccnv 5406  cfv 6188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196
This theorem is referenced by: (None)
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