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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrefsymrelsrel | Structured version Visualization version GIF version |
Description: For sets, being an element of the class of reflexive and symmetric relations is equivalent to satisfying the reflexive and symmetric relation predicates. (Contributed by Peter Mazsa, 23-Aug-2021.) |
Ref | Expression |
---|---|
elrefsymrelsrel | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3962 | . 2 ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ (𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels )) | |
2 | elrefrelsrel 37296 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅)) | |
3 | elsymrelsrel 37333 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅)) | |
4 | 2, 3 | anbi12d 632 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅))) |
5 | 1, 4 | bitrid 283 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∩ cin 3945 RefRels crefrels 36954 RefRel wrefrel 36955 SymRels csymrels 36960 SymRel wsymrel 36961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-rels 37261 df-ssr 37274 df-refs 37286 df-refrels 37287 df-refrel 37288 df-syms 37318 df-symrels 37319 df-symrel 37320 |
This theorem is referenced by: (None) |
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