Theorem ec1cnvres 39399

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 39398	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ [𝐵]◡(𝑅 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
2	1	eqabdv 2897	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)})
3		df-rab 3417	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)}
4	2, 3	eqtr4di 2816	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  {crab 3416   class class class wbr 5106  ^◡ccnv 5654   ↾ cres 5657  [cec 8740

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-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  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-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744

This theorem is used by:  dfpred4  39602