Theorem elrels5 38445

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 38443	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2		dfrel5 38302	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
3	1, 2	bitrdi 287	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  dom cdm 5700   ↾ cres 5702  Rel wrel 5705   Rels crels 38137

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-rels 38441

This theorem is referenced by: (None)