Theorem eccnvepres 38668

Description: Restricted converse epsilon coset of 𝐵. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.)

Assertion

Ref	Expression
eccnvepres	⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴})

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

Proof of Theorem eccnvepres

Step	Hyp	Ref	Expression
1		brcnvep 38652	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡ E 𝑥 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1cd 642	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥) ↔ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)))
3	2	abbidv 2807	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)})
4		ecres 38667	. 2 ⊢ [𝐵](◡ E ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)}
5		df-rab 3394	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)}
6	3, 4, 5	3eqtr4g 2801	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴})

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  {crab 3393   class class class wbr 5075   E cep 5520  ^◡ccnv 5620   ↾ cres 5623  [cec 8635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8639

This theorem is referenced by: (None)