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Theorem elrefsymrels2 37060
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 37079) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 37004, cf. the comment of dfrefrels2 37004. (Contributed by Peter Mazsa, 22-Jul-2019.)
Assertion
Ref Expression
elrefsymrels2 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))

Proof of Theorem elrefsymrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 refsymrels2 37056 . 2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
2 dmeq 5864 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
32reseq2d 5942 . . . 4 (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
53, 4sseq12d 3982 . . 3 (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6 cnveq 5834 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
76, 4sseq12d 3982 . . 3 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
85, 7anbi12d 632 . 2 (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅)))
91, 8rabeqel 36743 1 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  cin 3914  wss 3915   I cid 5535  ccnv 5637  dom cdm 5638  cres 5640   Rels crels 36665   RefRels crefrels 36668   SymRels csymrels 36674
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-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-rels 36976  df-ssr 36989  df-refs 37001  df-refrels 37002  df-syms 37033  df-symrels 37034
This theorem is referenced by:  elrefsymrels3  37061
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