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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 6047 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | |
2 | inss2 4229 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | |
3 | 1, 2 | eqsstrri 4017 | 1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3947 ⊆ wss 3948 I cid 5573 × cxp 5674 ↾ cres 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-res 5688 |
This theorem is referenced by: resiexg 7904 hartogslem1 9536 dfle2 13125 hausdiag 23148 qtophaus 32811 bj-imdirid 36062 bj-iminvid 36071 idresssidinxp 37172 rtrclex 42358 rtrclexi 42362 idhe 42528 |
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