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Theorem refsymrels2 34653
Description: Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~? dfeqvrels2 ) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 34605, cf. the comment of dfrefrels2 34605. (Contributed by Peter Mazsa, 20-Jul-2019.)
Assertion
Ref Expression
refsymrels2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}

Proof of Theorem refsymrels2
StepHypRef Expression
1 dfrefrels2 34605 . . 3 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dfsymrels2 34633 . . 3 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
31, 2ineq12i 3963 . 2 ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟})
4 inrab 4047 . 2 ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟}) = {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)}
5 symrefref2 34651 . . . 4 (𝑟𝑟 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ↾ dom 𝑟) ⊆ 𝑟))
65pm5.32ri 565 . . 3 ((( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟))
76rabbii 3335 . 2 {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟𝑟𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
83, 4, 73eqtri 2797 1 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  {crab 3065  cin 3722  wss 3723   I cid 5156   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251   Rels crels 34317   RefRels crefrels 34320   SymRels csymrels 34326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-fun 6033  df-fn 6034  df-rels 34577  df-ssr 34590  df-refs 34602  df-refrels 34603  df-syms 34630  df-symrels 34631
This theorem is referenced by:  refsymrels3  34654  elrefsymrels2  34657
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