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Theorem bj-evalval 37077
Description: Value of the evaluation at a class. (Closed form of strfvnd 17223 and strfvn 17224). (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.)
Assertion
Ref Expression
bj-evalval (𝐹𝑉 → (Slot 𝐴𝐹) = (𝐹𝐴))

Proof of Theorem bj-evalval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elex 3500 . 2 (𝐹𝑉𝐹 ∈ V)
2 fveq1 6904 . . 3 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
3 df-slot 17220 . . 3 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 fvex 6918 . . 3 (𝐹𝐴) ∈ V
52, 3, 4fvmpt 7015 . 2 (𝐹 ∈ V → (Slot 𝐴𝐹) = (𝐹𝐴))
61, 5syl 17 1 (𝐹𝑉 → (Slot 𝐴𝐹) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  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-pr 5431
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-ss 3967  df-nul 4333  df-if 4525  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-iota 6513  df-fun 6562  df-fv 6568  df-slot 17220
This theorem is referenced by:  bj-evalid  37078  bj-evalidval  37080
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