Theorem bj-diagval 37155

Description: Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 37156 views it as the diagonal function. See df-bj-diag 37154 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 37154	. 2 ⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
2		reseq2 5934	. 2 ⊢ (𝑥 = 𝐴 → ( I ↾ 𝑥) = ( I ↾ 𝐴))
3		elex 3465	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		resiexg 7868	. 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 1540   ∈ wcel 2109  Vcvv 3444   I cid 5525   ↾ cres 5633  ‘cfv 6499  Idcdiag2 37153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6452  df-fun 6501  df-fv 6507  df-bj-diag 37154

This theorem is referenced by:  bj-diagval2  37156