Theorem elcnvcnvlem 44353

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 6189	. . . 4 ⊢ ◡◡𝐵 = (𝐵 ∩ (V × V))
2		incom 4161	. . . 4 ⊢ (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵)
3	1, 2	eqtri 2785	. . 3 ⊢ ◡◡𝐵 = ((V × V) ∩ 𝐵)
4	3	eleq2i 2854	. 2 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ 𝐴 ∈ ((V × V) ∩ 𝐵))
5		elinlem 44352	. 2 ⊢ (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
6	4, 5	bitri 278	1 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∈ wcel 2142  Vcvv 3454   ∩ cin 3903   I cid 5554   × cxp 5658  ^◡ccnv 5659  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is used by: (None)