Theorem ec1cnvres 39011

Description: Converse restricted coset of 𝐵. (Contributed by Peter Mazsa, 22-Mar-2019.) (Revised by Peter Mazsa, 21-Oct-2021.)

Assertion

Ref	Expression
ec1cnvres	⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉

Proof of Theorem ec1cnvres

Step	Hyp	Ref	Expression
1		elec1cnvres 39010	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ [𝐵]◡(𝑅 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
2	1	eqabdv 2895	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)})
3		df-rab 3415	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)}
4	2, 3	eqtr4di 2815	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  {crab 3414   class class class wbr 5107  ^◡ccnv 5658   ↾ cres 5661  [cec 8697

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8701

This theorem is used by:  dfpred4  39214