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Theorem elcnvcnvlem 36724
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 5488 . . . 4 𝐵 = (𝐵 ∩ (V × V))
2 incom 3763 . . . 4 (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵)
31, 2eqtri 2628 . . 3 𝐵 = ((V × V) ∩ 𝐵)
43eleq2i 2676 . 2 (𝐴𝐵𝐴 ∈ ((V × V) ∩ 𝐵))
5 elinlem 36723 . 2 (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
64, 5bitri 262 1 (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wcel 1976  Vcvv 3169  cin 3535   I cid 4935   × cxp 5023  ccnv 5024  cfv 5787
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-sbc 3399  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-uni 4364  df-br 4575  df-opab 4635  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-iota 5751  df-fun 5789  df-fv 5795
This theorem is referenced by: (None)
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