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Theorem bj-diagval2 34609
 Description: Value of the funtionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 34608 views it as the functionalized identity. See df-bj-diag 34607 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 34608 . 2 (𝐴𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
2 idinxpresid 5883 . 2 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
31, 2eqtr4di 2851 1 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ∩ cin 3880   I cid 5425   × cxp 5518   ↾ cres 5522  ‘cfv 6325  Idcdiag2 34606 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-res 5532  df-iota 6284  df-fun 6327  df-fv 6333  df-bj-diag 34607 This theorem is referenced by:  bj-eldiag  34610  bj-eldiag2  34611
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