Theorem bj-evalid 37571

Description: The evaluation at a set of the identity function is that set. General form of ndxarg 17234. The restriction to a set 𝑉 is necessary since the argument of the function Slot 𝐴 (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.)

Assertion

Ref	Expression
bj-evalid	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Proof of Theorem bj-evalid

Step	Hyp	Ref	Expression
1		resiexg 7895	. . 3 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑉) ∈ V)
2		bj-evalval 37570	. . 3 ⊢ (( I ↾ 𝑉) ∈ V → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴))
3	1, 2	syl 17	. 2 ⊢ (𝑉 ∈ 𝑊 → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴))
4		fvresi 7159	. 2 ⊢ (𝐴 ∈ 𝑉 → (( I ↾ 𝑉)‘𝐴) = 𝐴)
5	3, 4	sylan9eq 2819	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456   I cid 5543   ↾ cres 5651  ‘cfv 6523  Slot cslot 17219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-res 5661  df-iota 6479  df-fun 6525  df-fv 6531  df-slot 17220

This theorem is referenced by:  bj-ndxarg  37572  bj-evalidval  37573