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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosselrels | Structured version Visualization version GIF version |
Description: Cosets of sets are elements of the relations class. Implies ⊢ (𝑅 ∈ Rels → ≀ 𝑅 ∈ Rels ). (Contributed by Peter Mazsa, 25-Aug-2021.) |
Ref | Expression |
---|---|
cosselrels | ⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ Rels ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossex 37745 | . 2 ⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V) | |
2 | relcoss 37749 | . . 3 ⊢ Rel ≀ 𝐴 | |
3 | elrelsrel 37813 | . . 3 ⊢ ( ≀ 𝐴 ∈ V → ( ≀ 𝐴 ∈ Rels ↔ Rel ≀ 𝐴)) | |
4 | 2, 3 | mpbiri 258 | . 2 ⊢ ( ≀ 𝐴 ∈ V → ≀ 𝐴 ∈ Rels ) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ Rels ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3466 Rel wrel 5671 ≀ ccoss 37499 Rels crels 37501 |
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-pow 5353 ax-pr 5417 ax-un 7718 |
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-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-coss 37737 df-rels 37811 |
This theorem is referenced by: cosscnvelrels 37823 dffunsALTV2 38010 dffunsALTV3 38011 dffunsALTV4 38012 elfunsALTV2 38019 elfunsALTV3 38020 elfunsALTV4 38021 elfunsALTV5 38022 |
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