Theorem eccnvepres2 39332

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		elecreseq 8774	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = [𝐵]◡ E )
2		eccnvep 39329	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]◡ E = 𝐵)
3	1, 2	eqtrd 2795	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   E cep 5550  ^◡ccnv 5650   ↾ cres 5653  [cec 8722

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8726

This theorem is used by:  eccnvepres3  39333  ecuncnvepres  39436