Theorem refsymrels3 36800

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 dfeqvrels3 36823) can use the ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) version of dfrefrels3 36748, cf. the comment of dfrefrel3 36750. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
refsymrels3	⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))}

Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem refsymrels3

Step	Hyp	Ref	Expression
1		refsymrels2 36799	. 2 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
2		idrefALT 6039	. . 3 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥)
3		cnvsym 6040	. . 3 ⊢ (◡𝑟 ⊆ 𝑟 ↔ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))
4	2, 3	anbi12i 627	. 2 ⊢ ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ↔ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)))
5	1, 4	rabbieq 36482	1 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))}

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1538   = wceq 1540  ∀wral 3062  {crab 3404   ∩ cin 3896   ⊆ wss 3897   class class class wbr 5087   I cid 5506  ^◡ccnv 5607  dom cdm 5608   ↾ cres 5610   Rels crels 36407   RefRels crefrels 36410   SymRels csymrels 36416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-br 5088  df-opab 5150  df-id 5507  df-xp 5614  df-rel 5615  df-cnv 5616  df-dm 5618  df-rn 5619  df-res 5620  df-rels 36719  df-ssr 36732  df-refs 36744  df-refrels 36745  df-syms 36776  df-symrels 36777

This theorem is referenced by: (None)