Theorem bj-diagval 37879

Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 37880 views it as the diagonal function. See df-bj-diag 37878 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 37878	. 2 ⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
2		reseq2 5975	. 2 ⊢ (𝑥 = 𝐴 → ( I ↾ 𝑥) = ( I ↾ 𝐴))
3		elex 3478	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		resiexg 7915	. 2 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
5	1, 2, 3, 4	fvmptd3 7017	1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457   I cid 5557   ↾ cres 5665  ‘cfv 6540  Idcdiag2 37877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-bj-diag 37878

This theorem is used by:  bj-diagval2  37880