Theorem elcnvintab 42656

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 42655	. 2 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))
3	1	elcnvlem 42655	. 2 ⊢ (𝐴 ∈ ◡𝑥 ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ 𝑥))
4	2, 3	elmapintab 42650	1 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ ◡𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538   ∈ wcel 2105  {cab 2708  Vcvv 3473  ⟨cop 4635  ∩ cint 4951   ↦ cmpt 5232   × cxp 5675  ^◡ccnv 5676  ‘cfv 6544  1^st c1st 7976  2^nd c2nd 7977

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7978  df-2nd 7979

This theorem is referenced by:  cnvintabd  42657