Theorem elrels6 38827

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

Assertion

Ref	Expression
elrels6	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅))

Proof of Theorem elrels6

Step	Hyp	Ref	Expression
1		elrelsrel 38824	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2		dfrel6 38729	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	1, 2	bitrdi 289	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121   ∩ cin 3884   × cxp 5619  dom cdm 5621  ran crn 5622  Rel wrel 5626   Rels crels 38567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-rels 38822

This theorem is referenced by:  dfrefrels2  38975  dfcnvrefrels2  38990  dfsymrels2  39007  dftrrels2  39041