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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressucdifsn2 | Structured version Visualization version GIF version |
Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38226. (Contributed by Peter Mazsa, 24-Jul-2024.) |
Ref | Expression |
---|---|
ressucdifsn2 | ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjcsn 9642 | . 2 ⊢ (𝐴 ∩ {𝐴}) = ∅ | |
2 | disjresundif 38224 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ → ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: ressucdifsn 38226 partsuc2 38761 |
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