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Theorem dfcnvrefrel2 38486
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
StepHypRef Expression
1 df-cnvrefrel 38483 . 2 ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 dfrel6 38303 . . . . 5 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32biimpi 216 . . . 4 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
43sseq1d 4040 . . 3 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
54pm5.32ri 575 . 2 (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
61, 5bitri 275 1 ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  cin 3975  wss 3976   I cid 5592   × cxp 5698  dom cdm 5700  ran crn 5701  Rel wrel 5705   CnvRefRel wcnvrefrel 38144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-cnvrefrel 38483
This theorem is referenced by:  elcnvrefrelsrel  38492  cnvrefrelcoss2  38493
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