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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 6068 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | |
2 | inss2 4246 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | |
3 | 1, 2 | eqsstrri 4031 | 1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3962 ⊆ wss 3963 I cid 5582 × cxp 5687 ↾ 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-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
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-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-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: resiexg 7935 hartogslem1 9580 dfle2 13186 hausdiag 23669 qtophaus 33797 bj-imdirid 37169 bj-iminvid 37178 idresssidinxp 38290 rtrclex 43607 rtrclexi 43611 idhe 43777 |
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