Theorem bj-diagval2 36710

Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 36709 views it as the functionalized identity. See df-bj-diag 36708 for the terminology. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-diagval2	⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Proof of Theorem bj-diagval2

Step	Hyp	Ref	Expression
1		bj-diagval 36709	. 2 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
2		idinxpresid 6046	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
3	1, 2	eqtr4di 2783	1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∩ cin 3939   I cid 5569   × cxp 5670   ↾ cres 5674  ‘cfv 6542  Idcdiag2 36707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-bj-diag 36708

This theorem is referenced by:  bj-eldiag  36711  bj-eldiag2  36712