Theorem idinxpresid 6009

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 6008	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ (𝐴 ∩ 𝐴))
2		inidm 4168	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	reseq2i 5937	. 2 ⊢ ( I ↾ (𝐴 ∩ 𝐴)) = ( I ↾ 𝐴)
4	1, 3	eqtri 2760	1 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)

Syntax hints:   = wceq 1542   ∩ cin 3889   I cid 5520   × cxp 5624   ↾ cres 5628

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 5232  ax-pr 5372

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 5521  df-xp 5632  df-rel 5633  df-res 5638

This theorem is referenced by:  idssxp  6010  bj-diagval2  37509  idinxpssinxp2  38663