Theorem refrelsredund4 37992

Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 37873) 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 5683	. . . . 5 ⊢ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ↾ dom 𝑟)
2		sstr2 3981	. . . . 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 37607	. . 3 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
5		dfrefrels2 37873	. . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
6	4, 5	sseqtrri 4011	. 2 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ RefRels
7		in32 4213	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels )
8		inrab 4298	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
9		dfsymrels2 37905	. . . . . . 7 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
10	9	ineq2i 4201	. . . . . 6 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟})
11		refsymrels2 37925	. . . . . 6 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
12	8, 10, 11	3eqtr4i 2762	. . . . 5 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ( RefRels ∩ SymRels )
13	12	ineq1i 4200	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ SymRels ) ∩ RefRels )
14		inass 4211	. . . 4 ⊢ (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels ))
15	7, 13, 14	3eqtr3ri 2761	. . 3 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = (( RefRels ∩ SymRels ) ∩ RefRels )
16		in32 4213	. . 3 ⊢ (( RefRels ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ RefRels ) ∩ SymRels )
17		inass 4211	. . 3 ⊢ (( RefRels ∩ RefRels ) ∩ SymRels ) = ( RefRels ∩ ( RefRels ∩ SymRels ))
18	15, 16, 17	3eqtri 2756	. 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = ( RefRels ∩ ( RefRels ∩ SymRels ))
19		df-redund 37984	. 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 708	1 ⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533  {crab 3424   ∩ cin 3939   ⊆ wss 3940   I cid 5563   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668   Rels crels 37535   RefRels crefrels 37538   SymRels csymrels 37544   Redund wredund 37554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-rels 37845  df-ssr 37858  df-refs 37870  df-refrels 37871  df-syms 37902  df-symrels 37903  df-redund 37984

This theorem is referenced by:  refrelsredund2  37993