Theorem elrefsymrels2 38587

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 38606) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 38531, cf. the comment of dfrefrels2 38531. (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 38583	. 2 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
2		dmeq 5883	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3	2	reseq2d 5966	. . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4		id 22	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
5	3, 4	sseq12d 3992	. . 3 ⊢ (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6		cnveq 5853	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
7	6, 4	sseq12d 3992	. . 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 38272	1 ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∩ cin 3925   ⊆ wss 3926   I cid 5547  ^◡ccnv 5653  dom cdm 5654   ↾ cres 5656   Rels crels 38201   RefRels crefrels 38204   SymRels csymrels 38210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-rels 38503  df-ssr 38516  df-refs 38528  df-refrels 38529  df-syms 38560  df-symrels 38561

This theorem is referenced by:  elrefsymrels3  38588