Theorem bj-diagval 38241

Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 38242 views it as the diagonal function. See df-bj-diag 38240 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 38240	. 2 ⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
2		reseq2 5970	. 2 ⊢ (𝑥 = 𝐴 → ( I ↾ 𝑥) = ( I ↾ 𝐴))
3		elex 3475	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		resiexg 7934	. 2 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V)
5	1, 2, 3, 4	fvmptd3 7028	1 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  Vcvv 3454   I cid 5549   ↾ cres 5657  ‘cfv 6547  Idcdiag2 38239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-iota 6501  df-fun 6549  df-fv 6555  df-bj-diag 38240

This theorem is used by:  bj-diagval2  38242