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Theorem dfcnvrefrels3 37037
Description: Alternate definition of the class of converse reflexive relations. (Contributed by Peter Mazsa, 22-Jul-2019.)
Assertion
Ref Expression
dfcnvrefrels3 CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfcnvrefrels3
StepHypRef Expression
1 df-cnvrefrels 37034 . . 3 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 37033 . . 3 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
31, 2abeqin 36758 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4 dmexg 7841 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
54elv 3450 . . . . 5 dom 𝑟 ∈ V
6 rnexg 7842 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
76elv 3450 . . . . 5 ran 𝑟 ∈ V
85, 7xpex 7688 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
9 inex2g 5278 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10 brcnvssr 37014 . . . 4 (( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
118, 9, 10mp2b 10 . . 3 (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))
12 inxpssidinxp 36823 . . 3 ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
1311, 12bitri 275 . 2 (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
143, 13rabbieq 36756 1 CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wral 3061  {crab 3406  Vcvv 3444  cin 3910  wss 3911   class class class wbr 5106   I cid 5531   × cxp 5632  ccnv 5633  dom cdm 5634  ran crn 5635   Rels crels 36682   S cssr 36683   CnvRefs ccnvrefs 36687   CnvRefRels ccnvrefrels 36688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-ssr 37006  df-cnvrefs 37033  df-cnvrefrels 37034
This theorem is referenced by:  elcnvrefrels3  37043
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