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Mirrors > Home > MPE Home > Th. List > Mathboxes > partsuc2 | Structured version Visualization version GIF version |
Description: Property of the partition. (Contributed by Peter Mazsa, 24-Jul-2024.) |
Ref | Expression |
---|---|
partsuc2 | ⊢ (((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) Part ((𝐴 ∪ {𝐴}) ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressucdifsn2 38148 | . 2 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) | |
2 | sucdifsn2 38142 | . 2 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴 | |
3 | parteq12 38681 | . 2 ⊢ ((((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) ∧ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴) → (((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) Part ((𝐴 ∪ {𝐴}) ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴)) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) Part ((𝐴 ∪ {𝐴}) ∖ {𝐴}) ↔ (𝑅 ↾ 𝐴) Part 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∖ cdif 3967 ∪ cun 3968 {csn 4648 ↾ cres 5701 Part wpart 38123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-reg 9657 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-ec 8761 df-qs 8765 df-coss 38316 df-cnvrefrel 38432 df-dmqs 38544 df-funALTV 38587 df-disjALTV 38610 df-part 38671 |
This theorem is referenced by: (None) |
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