![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > idssxp | Structured version Visualization version GIF version |
Description: A diagonal set as a subset of a Cartesian product. (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 | idinxpres 5672 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | |
2 | inss2 4029 | . 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | |
3 | 1, 2 | eqsstr3i 3832 | 1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3768 ⊆ wss 3769 I cid 5219 × cxp 5310 ↾ cres 5314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-res 5324 |
This theorem is referenced by: resiexg 7337 hausdiag 21777 qtophaus 30419 idresssidinxp 34574 |
Copyright terms: Public domain | W3C validator |