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Theorem dfcnvrefrels3 38227
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 38224 . . 3 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 38223 . . 3 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
31, 2abeqin 37950 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4 dmexg 7914 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
54elv 3468 . . . . 5 dom 𝑟 ∈ V
6 rnexg 7915 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
76elv 3468 . . . . 5 ran 𝑟 ∈ V
85, 7xpex 7761 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
9 inex2g 5325 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10 brcnvssr 38204 . . . 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 38014 . . 3 ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
1311, 12bitri 274 . 2 (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
143, 13rabbieq 3428 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 3051  {crab 3419  Vcvv 3462  cin 3946  wss 3947   class class class wbr 5153   I cid 5579   × cxp 5680  ccnv 5681  dom cdm 5682  ran crn 5683   Rels crels 37878   S cssr 37879   CnvRefs ccnvrefs 37883   CnvRefRels ccnvrefrels 37884
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 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-ssr 38196  df-cnvrefs 38223  df-cnvrefrels 38224
This theorem is referenced by:  elcnvrefrels3  38233
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