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Theorem elrefsymrels2 34676
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 34693) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 34624, cf. the comment of dfrefrels2 34624. (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 34672 . 2 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
2 dmeq 5492 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
32reseq2d 5565 . . . 4 (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
53, 4sseq12d 3794 . . 3 (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6 cnveq 5464 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
76, 4sseq12d 3794 . . 3 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
85, 7anbi12d 624 . 2 (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅)))
91, 8rabeqel 34386 1 (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  cin 3731  wss 3732   I cid 5184  ccnv 5276  dom cdm 5277  cres 5279   Rels crels 34338   RefRels crefrels 34341   SymRels csymrels 34347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-rels 34596  df-ssr 34609  df-refs 34621  df-refrels 34622  df-syms 34649  df-symrels 34650
This theorem is referenced by:  elrefsymrels3  34677
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