Theorem elcnvintab 44794

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 2763	. . 3 ⊢ (𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩) = (𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)
2	1	elcnvlem 44793	. 2 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))
3	1	elcnvlem 44793	. 2 ⊢ (𝐴 ∈ ◡𝑥 ↔ (𝐴 ∈ (V × V) ∧ ((𝑦 ∈ (V × V) ↦ ⟨(2nd ‘𝑦), (1st ‘𝑦)⟩)‘𝐴) ∈ 𝑥))
4	2, 3	elmapintab 44788	1 ⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ ◡𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2147  {cab 2741  Vcvv 3454  ⟨cop 4838  ∩ cint 4910   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654  ‘cfv 6558  1^st c1st 8023  2^nd c2nd 8024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-1st 8025  df-2nd 8026

This theorem is used by:  cnvintabd  44795