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Theorem bj-evalid 33986
Description: The evaluation at a set of the identity function is that set. (General form of ndxarg 16341.) 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 7482 . . 3 (𝑉𝑊 → ( I ↾ 𝑉) ∈ V)
2 bj-evalval 33985 . . 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 6805 . 2 (𝐴𝑉 → (( I ↾ 𝑉)‘𝐴) = 𝐴)
53, 4sylan9eq 2853 1 ((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  Vcvv 3440   I cid 5354  cres 5452  cfv 6232  Slot cslot 16315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-res 5462  df-iota 6196  df-fun 6234  df-fv 6240  df-slot 16320
This theorem is referenced by:  bj-ndxarg  33987  bj-evalidval  33989
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