Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idssxp Structured version   Visualization version   GIF version

Theorem idssxp 29270
 Description: A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.)
Assertion
Ref Expression
idssxp ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Proof of Theorem idssxp
StepHypRef Expression
1 fnresi 5966 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 fnrel 5947 . . 3 (( I ↾ 𝐴) Fn 𝐴 → Rel ( I ↾ 𝐴))
3 relssdmrn 5615 . . 3 (Rel ( I ↾ 𝐴) → ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)))
41, 2, 3mp2b 10 . 2 ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴))
5 dmresi 5416 . . 3 dom ( I ↾ 𝐴) = 𝐴
6 rnresi 5438 . . 3 ran ( I ↾ 𝐴) = 𝐴
75, 6xpeq12i 5097 . 2 (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) = (𝐴 × 𝐴)
84, 7sseqtri 3616 1 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3555   I cid 4984   × cxp 5072  dom cdm 5074  ran crn 5075   ↾ cres 5076  Rel wrel 5079   Fn wfn 5842 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-fun 5849  df-fn 5850 This theorem is referenced by:  qtophaus  29682
 Copyright terms: Public domain W3C validator