Theorem bj-diagval 37433

Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 37434 views it as the diagonal function. See df-bj-diag 37432 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 37432	. 2 ⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
2		reseq2 5941	. 2 ⊢ (𝑥 = 𝐴 → ( I ↾ 𝑥) = ( I ↾ 𝐴))
3		elex 3463	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		resiexg 7864	. 2 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
5	1, 2, 3, 4	fvmptd3 6973	1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   I cid 5526   ↾ cres 5634  ‘cfv 6500  Idcdiag2 37431

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-bj-diag 37432

This theorem is referenced by:  bj-diagval2  37434