Theorem elcnvrefrels2 39349

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 39343	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
2		id 23	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		dmeq 5891	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5924	. . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	xpeq12d 5690	. . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
6	5	ineq2d 4169	. . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
7	2, 6	sseq12d 3967	. 2 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
8	1, 7	rabeqel 38992	1 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3901   ⊆ wss 3902   I cid 5553   × cxp 5657  dom cdm 5659  ran crn 5660   Rels crels 38920   CnvRefRels ccnvrefrels 38926

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 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-rels 39175  df-ssr 39313  df-cnvrefs 39340  df-cnvrefrels 39341

This theorem is used by:  elcnvrefrelsrel  39351  cosselcnvrefrels2  39353