Theorem eccnvepres2 38279

Description: The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019.)

Assertion

Ref	Expression
eccnvepres2	⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵)

Proof of Theorem eccnvepres2

Step	Hyp	Ref	Expression
1		ecres2 38273	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = [𝐵]◡ E )
2		eccnvep 38276	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]◡ E = 𝐵)
3	1, 2	eqtrd 2776	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2107   E cep 5589  ^◡ccnv 5689   ↾ cres 5692  [cec 8748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150  df-opab 5212  df-eprel 5590  df-xp 5696  df-rel 5697  df-cnv 5698  df-dm 5700  df-rn 5701  df-res 5702  df-ima 5703  df-ec 8752

This theorem is referenced by:  eccnvepres3  38280