Theorem refsymrels2 38667

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 38690) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38611, cf. the comment of dfrefrels2 38611. (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 38611	. . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2		dfsymrels2 38643	. . 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 38665	. . . 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 3400	. 2 ⊢ {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
8	3, 4, 7	3eqtri 2758	1 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}

Syntax hints:   ∧ wa 395   = wceq 1541  {crab 3395   ∩ cin 3896   ⊆ wss 3897   I cid 5513   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   Rels crels 38230   RefRels crefrels 38233   SymRels csymrels 38239

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 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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  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 38470  df-ssr 38596  df-refs 38608  df-refrels 38609  df-syms 38640  df-symrels 38641

This theorem is referenced by:  refsymrels3  38668  elrefsymrels2  38671  dfeqvrels2  38690  refrelsredund4  38734