Theorem refsymrels2 36606

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 36628) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 36558, cf. the comment of dfrefrels2 36558. (Contributed by Peter Mazsa, 20-Jul-2019.)

Assertion

Ref	Expression
refsymrels2	⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}

Proof of Theorem refsymrels2

Step	Hyp	Ref	Expression
1		dfrefrels2 36558	. . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2		dfsymrels2 36586	. . 3 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
3	1, 2	ineq12i 4141	. 2 ⊢ ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟})
4		inrab 4237	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
5		symrefref2 36604	. . . 4 ⊢ (◡𝑟 ⊆ 𝑟 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ↾ dom 𝑟) ⊆ 𝑟))
6	5	pm5.32ri 575	. . 3 ⊢ ((( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ↔ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟))
7	6	rabbii 3397	. 2 ⊢ {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
8	3, 4, 7	3eqtri 2770	1 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}

Syntax hints:   ∧ wa 395   = wceq 1539  {crab 3067   ∩ cin 3882   ⊆ wss 3883   I cid 5479   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   Rels crels 36262   RefRels crefrels 36265   SymRels csymrels 36271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-rels 36530  df-ssr 36543  df-refs 36555  df-refrels 36556  df-syms 36583  df-symrels 36584

This theorem is referenced by:  refsymrels3  36607  elrefsymrels2  36610  dfeqvrels2  36628  refrelsredund4  36672