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| Mirrors > Home > MPE Home > Th. List > Mathboxes > partsuc | Structured version Visualization version GIF version | ||
| Description: Property of the partition. (Contributed by Peter Mazsa, 20-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| partsuc | ⊢ (((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) Part (suc 𝐴 ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ressucdifsn 38247 | . 2 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) | |
| 2 | sucdifsn 38241 | . 2 ⊢ (suc 𝐴 ∖ {𝐴}) = 𝐴 | |
| 3 | parteq12 38778 | . 2 ⊢ ((((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) ∧ (suc 𝐴 ∖ {𝐴}) = 𝐴) → (((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) Part (suc 𝐴 ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) Part (suc 𝐴 ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∖ cdif 3947 {csn 4625 ↾ cres 5686 suc csuc 6385 Part wpart 38222 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-reg 9633 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-suc 6389 df-ec 8748 df-qs 8752 df-coss 38413 df-cnvrefrel 38529 df-dmqs 38641 df-funALTV 38684 df-disjALTV 38707 df-part 38768 | 
| This theorem is referenced by: (None) | 
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