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| Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelressn | Structured version Visualization version GIF version | ||
| Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38443) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.) |
| Ref | Expression |
|---|---|
| refrelressn | ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | refressn 38444 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥) | |
| 2 | relres 6023 | . 2 ⊢ Rel (𝑅 ↾ {𝐴}) | |
| 3 | dfrefrel5 38518 | . 2 ⊢ ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴}))) | |
| 4 | 1, 2, 3 | sylanblrc 590 | 1 ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 {csn 4626 class class class wbr 5143 dom cdm 5685 ran crn 5686 ↾ cres 5687 Rel wrel 5690 RefRel wrefrel 38188 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-refrel 38513 |
| This theorem is referenced by: (None) |
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