Theorem eccnvepres3 38627

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 6190	. . 3 ⊢ (◡ E ↾ dom (◡ E ↾ 𝐴)) = (◡ E ↾ 𝐴)
2	1	eceq2i 8679	. 2 ⊢ [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = [𝐵](◡ E ↾ 𝐴)
3		eccnvepres2 38626	. 2 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = 𝐵)
4	2, 3	eqtr3id 2786	1 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   E cep 5523  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626  [cec 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8638

This theorem is referenced by:  eldisjlem19  39248