Theorem eccnvepres3 37667

Description: Condition for a restricted converse epsilon coset of a set to be the set itself. (Contributed by Peter Mazsa, 11-May-2021.)

Assertion

Ref	Expression
eccnvepres3	⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ 𝐴) = 𝐵)

Proof of Theorem eccnvepres3

Step	Hyp	Ref	Expression
1		resdmres 6225	. . 3 ⊢ (◡ E ↾ dom (◡ E ↾ 𝐴)) = (◡ E ↾ 𝐴)
2	1	eceq2i 8746	. 2 ⊢ [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = [𝐵](◡ E ↾ 𝐴)
3		eccnvepres2 37666	. 2 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = 𝐵)
4	2, 3	eqtr3id 2780	1 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   E cep 5572  ^◡ccnv 5668  dom cdm 5669   ↾ cres 5671  [cec 8703

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

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 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707

This theorem is referenced by:  eldisjlem19  38193