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| Mirrors > Home > MPE Home > Th. List > idssxp | Structured version Visualization version GIF version | ||
| Description: A diagonal set as a subset of a Cartesian square. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| idssxp | ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idinxpresid 6065 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | |
| 2 | inss2 4237 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | |
| 3 | 1, 2 | eqsstrri 4030 | 1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∩ cin 3949 ⊆ wss 3950 I cid 5576 × cxp 5682 ↾ cres 5686 | 
| 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-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-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-res 5696 | 
| This theorem is referenced by: resiexg 7935 hartogslem1 9583 dfle2 13190 hausdiag 23654 qtophaus 33836 bj-imdirid 37188 bj-iminvid 37197 idresssidinxp 38310 rtrclex 43635 rtrclexi 43639 idhe 43805 | 
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