Theorem ec1cnvres 38614

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 38613	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ [𝐵]◡(𝑅 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)))
2	1	eqabdv 2870	. 2 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)})
3		df-rab 3391	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵)}
4	2, 3	eqtr4di 2790	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵]◡(𝑅 ↾ 𝐴) = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  {crab 3390   class class class wbr 5086  ^◡ccnv 5624   ↾ cres 5627  [cec 8635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639

This theorem is referenced by:  dfpred4  38817