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Theorem bj-diagval2 37118
Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 37117 views it as the functionalized identity. See df-bj-diag 37116 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 37117 . 2 (𝐴𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
2 idinxpresid 6063 . 2 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
31, 2eqtr4di 2791 1 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1535  wcel 2104  cin 3962   I cid 5576   × cxp 5682  cres 5686  cfv 6559  Idcdiag2 37115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7748
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4916  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6511  df-fun 6561  df-fv 6567  df-bj-diag 37116
This theorem is referenced by:  bj-eldiag  37119  bj-eldiag2  37120
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