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Theorem idinxpresid 5902
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
StepHypRef Expression
1 idinxpres 5901 . 2 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ (𝐴𝐴))
2 inidm 4179 . . 3 (𝐴𝐴) = 𝐴
32reseq2i 5837 . 2 ( I ↾ (𝐴𝐴)) = ( I ↾ 𝐴)
41, 3eqtri 2847 1 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cin 3918   I cid 5446   × cxp 5540  cres 5544
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-res 5554
This theorem is referenced by:  idssxp  5903  bj-diagval2  34503  idinxpssinxp2  35645
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