Theorem eccnvepres 39219

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 39203	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡ E 𝑥 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1cd 647	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥) ↔ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)))
3	2	abbidv 2826	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)})
4		ecres 39218	. 2 ⊢ [𝐵](◡ E ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)}
5		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)}
6	3, 4, 5	3eqtr4g 2820	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  {crab 3412   class class class wbr 5102   E cep 5549  ^◡ccnv 5649   ↾ cres 5652  [cec 8715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8719

This theorem is used by: (None)