Theorem idinxpresid 6077

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 6076	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ (𝐴 ∩ 𝐴))
2		inidm 4248	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	reseq2i 6006	. 2 ⊢ ( I ↾ (𝐴 ∩ 𝐴)) = ( I ↾ 𝐴)
4	1, 3	eqtri 2768	1 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)

Syntax hints:   = wceq 1537   ∩ cin 3975   I cid 5592   × cxp 5698   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by:  idssxp  6078  bj-diagval2  37141  idinxpssinxp2  38274