Theorem eccnvepres3 38999

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 6235	. . 3 ⊢ (◡ E ↾ dom (◡ E ↾ 𝐴)) = (◡ E ↾ 𝐴)
2	1	eceq2i 8743	. 2 ⊢ [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = [𝐵](◡ E ↾ 𝐴)
3		eccnvepres2 38998	. 2 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = 𝐵)
4	2, 3	eqtr3id 2814	1 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ 𝐴) = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   E cep 5562  ^◡ccnv 5662  dom cdm 5663   ↾ cres 5665  [cec 8698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8702

This theorem is used by:  eldisjlem19  39620