Theorem ec1cnvres 38408

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 38407	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ [𝐵]◡(𝑅 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
2	1	eqabdv 2867	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)})
3		df-rab 3398	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)}
4	2, 3	eqtr4di 2787	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  {crab 3397   class class class wbr 5096  ^◡ccnv 5621   ↾ cres 5624  [cec 8631

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ec 8635

This theorem is referenced by:  dfpred4  38592