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Theorem bj-evalid 37078
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 ((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Proof of Theorem bj-evalid
StepHypRef Expression
1 resiexg 7935 . . 3 (𝑉𝑊 → ( I ↾ 𝑉) ∈ V)
2 bj-evalval 37077 . . 3 (( I ↾ 𝑉) ∈ V → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴))
31, 2syl 17 . 2 (𝑉𝑊 → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴))
4 fvresi 7194 . 2 (𝐴𝑉 → (( I ↾ 𝑉)‘𝐴) = 𝐴)
53, 4sylan9eq 2796 1 ((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479   I cid 5576  cres 5686  cfv 6560  Slot cslot 17219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-slot 17220
This theorem is referenced by:  bj-ndxarg  37079  bj-evalidval  37080
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