Theorem ec1cnvres 38814

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 38813	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ [𝐵]◡(𝑅 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
2	1	eqabdv 2902	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)})
3		df-rab 3424	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)}
4	2, 3	eqtr4di 2822	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  {crab 3423   class class class wbr 5113  ^◡ccnv 5661   ↾ cres 5664  [cec 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ec 8695

This theorem is referenced by:  dfpred4  39017