Theorem elcnvcnvlem 44044

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 6150	. . . 4 ⊢ ◡◡𝐵 = (𝐵 ∩ (V × V))
2		incom 4145	. . . 4 ⊢ (𝐵 ∩ (V × V)) = ((V × V) ∩ 𝐵)
3	1, 2	eqtri 2763	. . 3 ⊢ ◡◡𝐵 = ((V × V) ∩ 𝐵)
4	3	eleq2i 2832	. 2 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ 𝐴 ∈ ((V × V) ∩ 𝐵))
5		elinlem 44043	. 2 ⊢ (𝐴 ∈ ((V × V) ∩ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
6	4, 5	bitri 276	1 ⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  Vcvv 3432   ∩ cin 3889   I cid 5519   × cxp 5623  ^◡ccnv 5624  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by: (None)