Theorem elcnvrefrels2 38625

Description: Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019.)

Assertion

Ref	Expression
elcnvrefrels2	⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))

Proof of Theorem elcnvrefrels2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcnvrefrels2 38619	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
2		id 22	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		dmeq 5842	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5875	. . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	xpeq12d 5645	. . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
6	5	ineq2d 4167	. . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
7	2, 6	sseq12d 3963	. 2 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
8	1, 7	rabeqel 38290	1 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∩ cin 3896   ⊆ wss 3897   I cid 5508   × cxp 5612  dom cdm 5614  ran crn 5615   Rels crels 38223   CnvRefRels ccnvrefrels 38229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-rels 38463  df-ssr 38589  df-cnvrefs 38616  df-cnvrefrels 38617

This theorem is referenced by:  elcnvrefrelsrel  38627  cosselcnvrefrels2  38629