Theorem idssxp 5945

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 5944	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
2		inss2 4160	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
3	1, 2	eqsstrri 3952	1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Syntax hints:   ∩ cin 3882   ⊆ wss 3883   I cid 5479   × cxp 5578   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-res 5592

This theorem is referenced by:  resiexg  7735  hartogslem1  9231  dfle2  12810  hausdiag  22704  qtophaus  31688  bj-imdirid  35284  bj-iminvid  35293  idresssidinxp  36371  rtrclex  41114  rtrclexi  41118  idhe  41284