Theorem idssxp 6008

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 6007	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
2		inss2 4179	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
3	1, 2	eqsstrri 3970	1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Syntax hints:   ∩ cin 3889   ⊆ wss 3890   I cid 5518   × cxp 5622   ↾ cres 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-res 5636

This theorem is referenced by:  resiexg  7856  hartogslem1  9450  dfle2  13089  hausdiag  23620  qtophaus  33996  bj-imdirid  37516  bj-iminvid  37525  idresssidinxp  38649  rtrclex  44062  rtrclexi  44066  idhe  44232