Theorem elrefsymrels2 38612

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 38631) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 38556, cf. the comment of dfrefrels2 38556. (Contributed by Peter Mazsa, 22-Jul-2019.)

Assertion

Ref	Expression
elrefsymrels2	⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))

Proof of Theorem elrefsymrels2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		refsymrels2 38608	. 2 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
2		dmeq 5843	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3	2	reseq2d 5928	. . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4		id 22	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
5	3, 4	sseq12d 3968	. . 3 ⊢ (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6		cnveq 5813	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
7	6, 4	sseq12d 3968	. . 3 ⊢ (𝑟 = 𝑅 → (◡𝑟 ⊆ 𝑟 ↔ ◡𝑅 ⊆ 𝑅))
8	5, 7	anbi12d 632	. 2 ⊢ (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅)))
9	1, 8	rabeqel 38295	1 ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∩ cin 3901   ⊆ wss 3902   I cid 5510  ^◡ccnv 5615  dom cdm 5616   ↾ cres 5618   Rels crels 38223   RefRels crefrels 38226   SymRels csymrels 38232

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-rels 38528  df-ssr 38541  df-refs 38553  df-refrels 38554  df-syms 38585  df-symrels 38586

This theorem is referenced by:  elrefsymrels3  38613