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Theorem idssxp 6047
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 6046 . . 3 ( I ↾ 𝐴) Fn 𝐴
2 fnrel 6027 . . 3 (( I ↾ 𝐴) Fn 𝐴 → Rel ( I ↾ 𝐴))
3 relssdmrn 5694 . . 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 5492 . . 3 dom ( I ↾ 𝐴) = 𝐴
6 rnresi 5514 . . 3 ran ( I ↾ 𝐴) = 𝐴
75, 6xpeq12i 5171 . 2 (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) = (𝐴 × 𝐴)
84, 7sseqtri 3670 1 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wss 3607   I cid 5052   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  Rel wrel 5148   Fn wfn 5921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928  df-fn 5929
This theorem is referenced by:  qtophaus  30031  idresssidinxp  34220  idinxpres  34229
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