Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-diagval2 Structured version   Visualization version   GIF version

Theorem bj-diagval2 35858
Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 35857 views it as the functionalized identity. See df-bj-diag 35856 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 35857 . 2 (𝐴𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
2 idinxpresid 6037 . 2 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
31, 2eqtr4di 2789 1 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cin 3943   I cid 5566   × cxp 5667  cres 5671  cfv 6532  Idcdiag2 35855
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6484  df-fun 6534  df-fv 6540  df-bj-diag 35856
This theorem is referenced by:  bj-eldiag  35859  bj-eldiag2  35860
  Copyright terms: Public domain W3C validator