Theorem elrefsymrels3 39679

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 39698) 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 39619, cf. the comment of dfrefrel3 39621. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
elrefsymrels3	⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem elrefsymrels3

Step	Hyp	Ref	Expression
1		elrefsymrels2 39678	. 2 ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
2		idrefALT 6106	. . . 4 ⊢ (( I ↾ dom 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥)
3		cnvsym 6107	. . . 4 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
4	2, 3	anbi12i 640	. . 3 ⊢ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
5	4	anbi1i 636	. 2 ⊢ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
6	1, 5	bitri 278	1 ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089   class class class wbr 5102   I cid 5544  ^◡ccnv 5649  dom cdm 5650   ↾ cres 5652   Rels crels 39210   RefRels crefrels 39213   SymRels csymrels 39219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-rels 39465  df-ssr 39603  df-refs 39615  df-refrels 39616  df-syms 39647  df-symrels 39648

This theorem is used by: (None)