Theorem elcnvintab 43595

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 2729	. . 3 ⊢ (𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩) = (𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
2	1	elcnvlem 43594	. 2 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))
3	1	elcnvlem 43594	. 2 ⊢ (𝐴 ∈ ◡𝑥 ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ 𝑥))
4	2, 3	elmapintab 43589	1 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ ◡𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  {cab 2707  Vcvv 3436  ⟨cop 4583  ∩ cint 4896   ↦ cmpt 5173   × cxp 5617  ^◡ccnv 5618  ‘cfv 6482  1^st c1st 7922  2^nd c2nd 7923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fv 6490  df-1st 7924  df-2nd 7925

This theorem is referenced by:  cnvintabd  43596