Theorem refsymrels2 38681

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 38704) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38625, cf. the comment of dfrefrels2 38625. (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 38625	. . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2		dfsymrels2 38657	. . 3 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
3	1, 2	ineq12i 4167	. 2 ⊢ ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟})
4		inrab 4265	. 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 38679	. . . 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 3401	. 2 ⊢ {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
8	3, 4, 7	3eqtri 2760	1 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}

Syntax hints:   ∧ wa 395   = wceq 1541  {crab 3396   ∩ cin 3897   ⊆ wss 3898   I cid 5513   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   Rels crels 38244   RefRels crefrels 38247   SymRels csymrels 38253

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-rels 38484  df-ssr 38610  df-refs 38622  df-refrels 38623  df-syms 38654  df-symrels 38655

This theorem is referenced by:  refsymrels3  38682  elrefsymrels2  38685  dfeqvrels2  38704  refrelsredund4  38748