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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 37768) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.) |
Ref | Expression |
---|---|
refrelressn | ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refressn 37769 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥) | |
2 | relres 6000 | . 2 ⊢ Rel (𝑅 ↾ {𝐴}) | |
3 | dfrefrel5 37843 | . 2 ⊢ ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴}))) | |
4 | 1, 2, 3 | sylanblrc 589 | 1 ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∀wral 3053 ∩ cin 3939 {csn 4620 class class class wbr 5138 dom cdm 5666 ran crn 5667 ↾ cres 5668 Rel wrel 5671 RefRel wrefrel 37505 |
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-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-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-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-refrel 37838 |
This theorem is referenced by: (None) |
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