Theorem elcnvrefrels2 39291

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 39285	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
2		id 23	. . 3 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		dmeq 5893	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4		rneq 5926	. . . . 5 ⊢ (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
5	3, 4	xpeq12d 5692	. . . 4 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
6	5	ineq2d 4173	. . 3 ⊢ (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
7	2, 6	sseq12d 3970	. 2 ⊢ (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
8	1, 7	rabeqel 38934	1 ⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∩ cin 3904   ⊆ wss 3905   I cid 5555   × cxp 5659  dom cdm 5661  ran crn 5662   Rels crels 38862   CnvRefRels ccnvrefrels 38868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-rels 39117  df-ssr 39255  df-cnvrefs 39282  df-cnvrefrels 39283

This theorem is used by:  elcnvrefrelsrel  39293  cosselcnvrefrels2  39295