Theorem extep 39189

Description: Property of epsilon relation, see also extid 39216, extssr 39489 and the comment of df-ssr 39478. (Contributed by Peter Mazsa, 10-Jul-2019.)

Assertion

Ref	Expression
extep	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵))

Proof of Theorem extep

Step	Hyp	Ref	Expression
1		eccnvep 39188	. 2 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)
2		eccnvep 39188	. 2 ⊢ (𝐵 ∈ 𝑊 → [𝐵]◡ E = 𝐵)
3	1, 2	eqeqan12d 2775	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   E cep 5550  ^◡ccnv 5650  [cec 8699

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 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  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 8703

This theorem is used by: (None)