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Theorem idssxp 6160
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 6159 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 fnrel 6140 . . 3 (( I ↾ 𝐴) Fn 𝐴 → Rel ( I ↾ 𝐴))
3 relssdmrn 5811 . . 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 5609 . . 3 dom ( I ↾ 𝐴) = 𝐴
6 rnresi 5630 . . 3 ran ( I ↾ 𝐴) = 𝐴
75, 6xpeq12i 5290 . 2 (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) = (𝐴 × 𝐴)
84, 7sseqtri 3793 1 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3729   I cid 5170   × cxp 5261  dom cdm 5263  ran crn 5264  cres 5265  Rel wrel 5268   Fn wfn 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-br 4798  df-opab 4860  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-fun 6044  df-fn 6045
This theorem is referenced by:  qtophaus  30260  idresssidinxp  34437  idinxpres  34446
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