Theorem dfcnvrefrel2 37926

Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 24-Jul-2019.)

Assertion

Ref	Expression
dfcnvrefrel2	⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Proof of Theorem dfcnvrefrel2

Step	Hyp	Ref	Expression
1		df-cnvrefrel 37923	. 2 ⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		dfrel6 37742	. . . . 5 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	2	biimpi 215	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	3	sseq1d 4009	. . 3 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
5	4	pm5.32ri 575	. 2 ⊢ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
6	1, 5	bitri 275	1 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∩ cin 3943   ⊆ wss 3944   I cid 5569   × cxp 5670  dom cdm 5672  ran crn 5673  Rel wrel 5677   CnvRefRel wcnvrefrel 37579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-cnvrefrel 37923

This theorem is referenced by:  elcnvrefrelsrel  37932  cnvrefrelcoss2  37933