Theorem elcnvcnvintab 39935

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 6043	. . . 4 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V))
2		incom 4177	. . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
3	1, 2	eqtri 2844	. . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})
4	3	eleq2i 2904	. 2 ⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}))
5		elinintab 39928	. 2 ⊢ (𝐴 ∈ ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
6	4, 5	bitri 277	1 ⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   ∈ wcel 2110  {cab 2799  Vcvv 3494   ∩ cin 3934  ∩ cint 4868   × cxp 5547  ^◡ccnv 5548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-int 4869  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557

This theorem is referenced by:  cnvcnvintabd  39953