Theorem bj-diagval 37218

Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 37219 views it as the diagonal function. See df-bj-diag 37217 for the terminology. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-diagval	⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))

Proof of Theorem bj-diagval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-diag 37217	. 2 ⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
2		reseq2 5922	. 2 ⊢ (𝑥 = 𝐴 → ( I ↾ 𝑥) = ( I ↾ 𝐴))
3		elex 3457	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		resiexg 7842	. 2 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
5	1, 2, 3, 4	fvmptd3 6952	1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436   I cid 5508   ↾ cres 5616  ‘cfv 6481  Idcdiag2 37216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-res 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-bj-diag 37217

This theorem is referenced by:  bj-diagval2  37219