Theorem elcnvcnvintab 44234

Description: Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.)

Assertion

Ref	Expression
elcnvcnvintab	⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elcnvcnvintab

Step	Hyp	Ref	Expression
1		cnvcnv 6191	. . . 4 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V))
2		incom 4170	. . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
3	1, 2	eqtri 2792	. . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
4	3	eleq2i 2861	. 2 ⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}))
5		elinintab 44227	. 2 ⊢ (𝐴 ∈ ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
6	4, 5	bitri 278	1 ⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   ∈ wcel 2149  {cab 2747  Vcvv 3463   ∩ cin 3912  ∩ cint 4916   × cxp 5660  ^◡ccnv 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-int 4917  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670

This theorem is referenced by:  cnvcnvintabd  44252