Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcnvrefrels3 Structured version   Visualization version   GIF version

Theorem dfcnvrefrels3 34905
 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 34902 . . 3 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 34901 . . 3 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
31, 2abeqin 34652 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4 dmexg 7375 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
54elv 3402 . . . . 5 dom 𝑟 ∈ V
6 rnexg 7376 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
76elv 3402 . . . . 5 ran 𝑟 ∈ V
85, 7xpex 7240 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
9 inex2ALTV 34733 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10 brcnvssr 34884 . . . 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 34715 . . 3 ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
1311, 12bitri 267 . 2 (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
143, 13rabbieq 34650 1 CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601   ∈ wcel 2107  ∀wral 3090  {crab 3094  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792   class class class wbr 4886   I cid 5260   × cxp 5353  ◡ccnv 5354  dom cdm 5355  ran crn 5356   Rels crels 34608   S cssr 34609   CnvRefs ccnvrefs 34613   CnvRefRels ccnvrefrels 34614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-ssr 34876  df-cnvrefs 34901  df-cnvrefrels 34902 This theorem is referenced by:  elcnvrefrels3  34909
 Copyright terms: Public domain W3C validator