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Theorem idssxp 28217
Description: A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.)
Assertion
Ref Expression
idssxp  |-  (  _I  |`  A )  C_  ( A  X.  A )

Proof of Theorem idssxp
StepHypRef Expression
1 fnresi 5708 . . 3  |-  (  _I  |`  A )  Fn  A
2 fnrel 5689 . . 3  |-  ( (  _I  |`  A )  Fn  A  ->  Rel  (  _I  |`  A ) )
3 relssdmrn 5372 . . 3  |-  ( Rel  (  _I  |`  A )  ->  (  _I  |`  A ) 
C_  ( dom  (  _I  |`  A )  X. 
ran  (  _I  |`  A ) ) )
41, 2, 3mp2b 10 . 2  |-  (  _I  |`  A )  C_  ( dom  (  _I  |`  A )  X.  ran  (  _I  |`  A ) )
5 dmresi 5176 . . 3  |-  dom  (  _I  |`  A )  =  A
6 rnresi 5197 . . 3  |-  ran  (  _I  |`  A )  =  A
75, 6xpeq12i 4872 . 2  |-  ( dom  (  _I  |`  A )  X.  ran  (  _I  |`  A ) )  =  ( A  X.  A
)
84, 7sseqtri 3496 1  |-  (  _I  |`  A )  C_  ( A  X.  A )
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3436    _I cid 4760    X. cxp 4848   dom cdm 4850   ran crn 4851    |` cres 4852   Rel wrel 4855    Fn wfn 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-fun 5600  df-fn 5601
This theorem is referenced by:  qtophaus  28659
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