Theorem refrelsredund4 39365

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 39242) 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 5678	. . . . 5 ⊢ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ↾ dom 𝑟)
2		sstr2 3944	. . . . 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 38901	. . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
5		dfrefrels2 39242	. . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
6	4, 5	sseqtrri 3986	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ RefRels
7		in32 4182	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels )
8		inrab 4269	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
9		dfsymrels2 39274	. . . . . . 7 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
10	9	ineq2i 4170	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟})
11		refsymrels2 39298	. . . . . 6 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
12	8, 10, 11	3eqtr4i 2796	. . . . 5 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ( RefRels ∩ SymRels )
13	12	ineq1i 4169	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ SymRels ) ∩ RefRels )
14		inass 4180	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels ))
15	7, 13, 14	3eqtr3ri 2795	. . 3 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = (( RefRels ∩ SymRels ) ∩ RefRels )
16		in32 4182	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ RefRels ) ∩ SymRels )
17		inass 4180	. . 3 ⊢ (( RefRels ∩ RefRels ) ∩ SymRels ) = ( RefRels ∩ ( RefRels ∩ SymRels ))
18	15, 16, 17	3eqtri 2790	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = ( RefRels ∩ ( RefRels ∩ SymRels ))
19		df-redund 39357	. 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 723	1 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  {crab 3416   ∩ cin 3904   ⊆ wss 3905   I cid 5555   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   Rels crels 38834   RefRels crefrels 38837   SymRels csymrels 38843   Redund wredund 38853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-rels 39089  df-ssr 39227  df-refs 39239  df-refrels 39240  df-syms 39271  df-symrels 39272  df-redund 39357

This theorem is referenced by:  refrelsredund2  39366