Theorem idssxp 5998

Description: A diagonal set as a subset of a Cartesian square. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.)

Assertion

Ref	Expression
idssxp	⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Proof of Theorem idssxp

Step	Hyp	Ref	Expression
1		idinxpresid 5997	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
2		inss2 4188	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
3	1, 2	eqsstrri 3982	1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Syntax hints:   ∩ cin 3901   ⊆ wss 3902   I cid 5510   × cxp 5614   ↾ cres 5618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-res 5628

This theorem is referenced by:  resiexg  7842  hartogslem1  9428  dfle2  13046  hausdiag  23561  qtophaus  33847  bj-imdirid  37226  bj-iminvid  37235  idresssidinxp  38348  rtrclex  43656  rtrclexi  43660  idhe  43826