Theorem idinxpresid 6036

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 6035	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ (𝐴 ∩ 𝐴))
2		inidm 4213	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	reseq2i 5969	. 2 ⊢ ( I ↾ (𝐴 ∩ 𝐴)) = ( I ↾ 𝐴)
4	1, 3	eqtri 2759	1 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)

Syntax hints:   = wceq 1541   ∩ cin 3942   I cid 5565   × cxp 5666   ↾ cres 5670

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5141  df-opab 5203  df-id 5566  df-xp 5674  df-rel 5675  df-res 5680

This theorem is referenced by:  idssxp  6037  bj-diagval2  35846  idinxpssinxp2  36978