Theorem bj-evalid 34797

Description: The evaluation at a set of the identity function is that set. (General form of ndxarg 16571.) 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 7629	. . 3 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑉) ∈ V)
2		bj-evalval 34796	. . 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 6931	. 2 ⊢ (𝐴 ∈ 𝑉 → (( I ↾ 𝑉)‘𝐴) = 𝐴)
5	3, 4	sylan9eq 2813	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   I cid 5432   ↾ cres 5529  ‘cfv 6339  Slot cslot 16545

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-res 5539  df-iota 6298  df-fun 6341  df-fv 6347  df-slot 16550

This theorem is referenced by:  bj-ndxarg  34798  bj-evalidval  34799