Theorem eccnvepres 37638

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 37623	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡ E 𝑥 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1cd 633	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥) ↔ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)))
3	2	abbidv 2793	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)})
4		ecres 37636	. 2 ⊢ [𝐵](◡ E ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝑥)}
5		df-rab 3425	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)}
6	3, 4, 5	3eqtr4g 2789	1 ⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  {crab 3424   class class class wbr 5138   E cep 5569  ^◡ccnv 5665   ↾ cres 5668  [cec 8697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-eprel 5570  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8701

This theorem is referenced by: (None)