Theorem idinxpresid 5952

Description: The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) (Proof shortened by BJ, 23-Dec-2023.)

Assertion

Ref	Expression
idinxpresid	⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)

Proof of Theorem idinxpresid

Step	Hyp	Ref	Expression
1		idinxpres 5951	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ (𝐴 ∩ 𝐴))
2		inidm 4157	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	reseq2i 5885	. 2 ⊢ ( I ↾ (𝐴 ∩ 𝐴)) = ( I ↾ 𝐴)
4	1, 3	eqtri 2767	1 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)

Syntax hints:   = wceq 1541   ∩ cin 3890   I cid 5487   × cxp 5586   ↾ cres 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-res 5600

This theorem is referenced by:  idssxp  5953  bj-diagval2  35325  idinxpssinxp2  36432