Theorem dfcnvrefrel2 38253

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 38250	. 2 ⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		dfrel6 38070	. . . . 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 4020	. . 3 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
5	4	pm5.32ri 574	. 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 274	1 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534   ∩ cin 3955   ⊆ wss 3956   I cid 5580   × cxp 5681  dom cdm 5683  ran crn 5684  Rel wrel 5688   CnvRefRel wcnvrefrel 37910

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 2102  ax-9 2110  ax-ext 2700  ax-sep 5304  ax-nul 5311  ax-pr 5434

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3428  df-v 3473  df-dif 3959  df-un 3961  df-in 3963  df-ss 3973  df-nul 4333  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5689  df-rel 5690  df-cnv 5691  df-dm 5693  df-rn 5694  df-res 5695  df-cnvrefrel 38250

This theorem is referenced by:  elcnvrefrelsrel  38259  cnvrefrelcoss2  38260