Theorem refrelsredund4 38655

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 38536) if the relations are symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)

Assertion

Ref	Expression
refrelsredund4	⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩

Proof of Theorem refrelsredund4

Step	Hyp	Ref	Expression
1		inxpssres 5676	. . . . 5 ⊢ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ↾ dom 𝑟)
2		sstr2 3970	. . . . 5 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ↾ dom 𝑟) → (( I ↾ dom 𝑟) ⊆ 𝑟 → ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟))
3	1, 2	ax-mp 5	. . . 4 ⊢ (( I ↾ dom 𝑟) ⊆ 𝑟 → ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟)
4	3	ssrabi 38273	. . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
5		dfrefrels2 38536	. . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
6	4, 5	sseqtrri 4013	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ RefRels
7		in32 4210	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels )
8		inrab 4296	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
9		dfsymrels2 38568	. . . . . . 7 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
10	9	ineq2i 4197	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟})
11		refsymrels2 38588	. . . . . 6 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
12	8, 10, 11	3eqtr4i 2769	. . . . 5 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ( RefRels ∩ SymRels )
13	12	ineq1i 4196	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ SymRels ) ∩ RefRels )
14		inass 4208	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels ))
15	7, 13, 14	3eqtr3ri 2768	. . 3 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = (( RefRels ∩ SymRels ) ∩ RefRels )
16		in32 4210	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ RefRels ) ∩ SymRels )
17		inass 4208	. . 3 ⊢ (( RefRels ∩ RefRels ) ∩ SymRels ) = ( RefRels ∩ ( RefRels ∩ SymRels ))
18	15, 16, 17	3eqtri 2763	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = ( RefRels ∩ ( RefRels ∩ SymRels ))
19		df-redund 38647	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩ ↔ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ RefRels ∧ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = ( RefRels ∩ ( RefRels ∩ SymRels ))))
20	6, 18, 19	mpbir2an 711	1 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  {crab 3420   ∩ cin 3930   ⊆ wss 3931   I cid 5552   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   Rels crels 38206   RefRels crefrels 38209   SymRels csymrels 38215   Redund wredund 38225

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-rels 38508  df-ssr 38521  df-refs 38533  df-refrels 38534  df-syms 38565  df-symrels 38566  df-redund 38647

This theorem is referenced by:  refrelsredund2  38656