Theorem elcnvintab 43694

Description: Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elcnvintab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩) = (𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
2	1	elcnvlem 43693	. 2 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))
3	1	elcnvlem 43693	. 2 ⊢ (𝐴 ∈ ◡𝑥 ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ 𝑥))
4	2, 3	elmapintab 43688	1 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ ◡𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   ∈ wcel 2111  {cab 2709  Vcvv 3436  ⟨cop 4579  ∩ cint 4895   ↦ cmpt 5170   × cxp 5612  ^◡ccnv 5613  ‘cfv 6481  1^st c1st 7919  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fv 6489  df-1st 7921  df-2nd 7922

This theorem is referenced by:  cnvintabd  43695