Theorem refrelsredund4 38751

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 38628) 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 5638	. . . . 5 ⊢ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ↾ dom 𝑟)
2		sstr2 3937	. . . . 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 38310	. . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
5		dfrefrels2 38628	. . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
6	4, 5	sseqtrri 3980	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ RefRels
7		in32 4179	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels )
8		inrab 4265	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
9		dfsymrels2 38660	. . . . . . 7 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
10	9	ineq2i 4166	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟})
11		refsymrels2 38684	. . . . . 6 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
12	8, 10, 11	3eqtr4i 2766	. . . . 5 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ( RefRels ∩ SymRels )
13	12	ineq1i 4165	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ SymRels ) ∩ RefRels )
14		inass 4177	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels ))
15	7, 13, 14	3eqtr3ri 2765	. . 3 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = (( RefRels ∩ SymRels ) ∩ RefRels )
16		in32 4179	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ RefRels ) ∩ SymRels )
17		inass 4177	. . 3 ⊢ (( RefRels ∩ RefRels ) ∩ SymRels ) = ( RefRels ∩ ( RefRels ∩ SymRels ))
18	15, 16, 17	3eqtri 2760	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = ( RefRels ∩ ( RefRels ∩ SymRels ))
19		df-redund 38743	. 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 1541  {crab 3396   ∩ cin 3897   ⊆ wss 3898   I cid 5515   × cxp 5619  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   ↾ cres 5623   Rels crels 38247   RefRels crefrels 38250   SymRels csymrels 38256   Redund wredund 38266

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 5238  ax-nul 5248  ax-pr 5374

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 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-rels 38487  df-ssr 38613  df-refs 38625  df-refrels 38626  df-syms 38657  df-symrels 38658  df-redund 38743

This theorem is referenced by:  refrelsredund2  38752