Theorem elrels5 36162

Description: Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.)

Assertion

Ref	Expression
elrels5	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅))

Proof of Theorem elrels5

Step	Hyp	Ref	Expression
1		elrelsrel 36160	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2		dfrel5 36036	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
3	1, 2	bitrdi 290	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539   ∈ wcel 2112  dom cdm 5525   ↾ cres 5527  Rel wrel 5530   Rels crels 35888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-rels 36158

This theorem is referenced by: (None)