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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 6002 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | |
2 | inss2 4190 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | |
3 | 1, 2 | eqsstrri 3980 | 1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3910 ⊆ wss 3911 I cid 5531 × cxp 5632 ↾ cres 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-res 5646 |
This theorem is referenced by: resiexg 7852 hartogslem1 9479 dfle2 13067 hausdiag 22999 qtophaus 32420 bj-imdirid 35660 bj-iminvid 35669 idresssidinxp 36772 rtrclex 41896 rtrclexi 41900 idhe 42066 |
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