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Theorem dfcnvrefrels3 37925
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 37922 . . 3 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 37921 . . 3 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
31, 2abeqin 37646 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4 dmexg 7901 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
54elv 3475 . . . . 5 dom 𝑟 ∈ V
6 rnexg 7902 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
76elv 3475 . . . . 5 ran 𝑟 ∈ V
85, 7xpex 7747 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
9 inex2g 5314 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10 brcnvssr 37902 . . . 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 37711 . . 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 37644 1 CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wral 3056  {crab 3427  Vcvv 3469  cin 3943  wss 3944   class class class wbr 5142   I cid 5569   × cxp 5670  ccnv 5671  dom cdm 5672  ran crn 5673   Rels crels 37572   S cssr 37573   CnvRefs ccnvrefs 37577   CnvRefRels ccnvrefrels 37578
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-pow 5359  ax-pr 5423  ax-un 7732
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-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-ssr 37894  df-cnvrefs 37921  df-cnvrefrels 37922
This theorem is referenced by:  elcnvrefrels3  37931
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