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Theorem refrelsredund4 37807
Description: The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 37688) 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
StepHypRef Expression
1 inxpssres 5694 . . . . 5 ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ↾ dom 𝑟)
2 sstr2 3990 . . . . 5 (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ↾ dom 𝑟) → (( I ↾ dom 𝑟) ⊆ 𝑟 → ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟))
31, 2ax-mp 5 . . . 4 (( I ↾ dom 𝑟) ⊆ 𝑟 → ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟)
43ssrabi 37422 . . 3 {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
5 dfrefrels2 37688 . . 3 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
64, 5sseqtrri 4020 . 2 {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ⊆ RefRels
7 in32 4222 . . . 4 (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels )
8 inrab 4307 . . . . . 6 ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟}) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
9 dfsymrels2 37720 . . . . . . 7 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
109ineq2i 4210 . . . . . 6 ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ 𝑟𝑟})
11 refsymrels2 37740 . . . . . 6 ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
128, 10, 113eqtr4i 2768 . . . . 5 ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) = ( RefRels ∩ SymRels )
1312ineq1i 4209 . . . 4 (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ SymRels ) ∩ RefRels )
14 inass 4220 . . . 4 (({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ RefRels ) ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels ))
157, 13, 143eqtr3ri 2767 . . 3 ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = (( RefRels ∩ SymRels ) ∩ RefRels )
16 in32 4222 . . 3 (( RefRels ∩ SymRels ) ∩ RefRels ) = (( RefRels ∩ RefRels ) ∩ SymRels )
17 inass 4220 . . 3 (( RefRels ∩ RefRels ) ∩ SymRels ) = ( RefRels ∩ ( RefRels ∩ SymRels ))
1815, 16, 173eqtri 2762 . 2 ({𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} ∩ ( RefRels ∩ SymRels )) = ( RefRels ∩ ( RefRels ∩ SymRels ))
19 df-redund 37799 . 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 ))))
206, 18, 19mpbir2an 707 1 {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  {crab 3430  cin 3948  wss 3949   I cid 5574   × cxp 5675  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679   Rels crels 37350   RefRels crefrels 37353   SymRels csymrels 37359   Redund wredund 37369
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-rels 37660  df-ssr 37673  df-refs 37685  df-refrels 37686  df-syms 37717  df-symrels 37718  df-redund 37799
This theorem is referenced by:  refrelsredund2  37808
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