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Theorem bj-diagval2 35081
Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 35080 views it as the functionalized identity. See df-bj-diag 35079 for the terminology. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-diagval2 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Proof of Theorem bj-diagval2
StepHypRef Expression
1 bj-diagval 35080 . 2 (𝐴𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
2 idinxpresid 5915 . 2 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
31, 2eqtr4di 2796 1 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  cin 3865   I cid 5454   × cxp 5549  cres 5553  cfv 6380  Idcdiag2 35078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-res 5563  df-iota 6338  df-fun 6382  df-fv 6388  df-bj-diag 35079
This theorem is referenced by:  bj-eldiag  35082  bj-eldiag2  35083
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