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Mirrors > Home > MPE Home > Th. List > Mathboxes > refsymrels2 | Structured version Visualization version GIF version |
Description: Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 35838) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 35768, cf. the comment of dfrefrels2 35768. (Contributed by Peter Mazsa, 20-Jul-2019.) |
Ref | Expression |
---|---|
refsymrels2 | ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrefrels2 35768 | . . 3 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} | |
2 | dfsymrels2 35796 | . . 3 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} | |
3 | 1, 2 | ineq12i 4187 | . 2 ⊢ ( RefRels ∩ SymRels ) = ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}) |
4 | inrab 4275 | . 2 ⊢ ({𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} ∩ {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}) = {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} | |
5 | symrefref2 35814 | . . . 4 ⊢ (◡𝑟 ⊆ 𝑟 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ↾ dom 𝑟) ⊆ 𝑟)) | |
6 | 5 | pm5.32ri 578 | . . 3 ⊢ ((( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟) ↔ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)) |
7 | 6 | rabbii 3473 | . 2 ⊢ {𝑟 ∈ Rels ∣ (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} |
8 | 3, 4, 7 | 3eqtri 2848 | 1 ⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 {crab 3142 ∩ cin 3935 ⊆ wss 3936 I cid 5459 × cxp 5553 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 Rels crels 35470 RefRels crefrels 35473 SymRels csymrels 35479 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-rels 35740 df-ssr 35753 df-refs 35765 df-refrels 35766 df-syms 35793 df-symrels 35794 |
This theorem is referenced by: refsymrels3 35817 elrefsymrels2 35820 dfeqvrels2 35838 refrelsredund4 35882 |
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