Theorem elcnvcnvlem 43588

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	⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))

Proof of Theorem elcnvcnvlem

Step	Hyp	Ref	Expression
1		cnvcnv 6213	. . . 4 ⊢ ◡◡𝐵 = (𝐵 ∩ (V × V))
2		incom 4216	. . . 4 ⊢ (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵)
3	1, 2	eqtri 2762	. . 3 ⊢ ◡◡𝐵 = ((V × V) ∩ 𝐵)
4	3	eleq2i 2830	. 2 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ 𝐴 ∈ ((V × V) ∩ 𝐵))
5		elinlem 43587	. 2 ⊢ (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
6	4, 5	bitri 275	1 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2105  Vcvv 3477   ∩ cin 3961   I cid 5581   × cxp 5686  ^◡ccnv 5687  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570

This theorem is referenced by: (None)