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Theorem bj-evalval 37058
Description: Value of the evaluation at a class. (Closed form of strfvnd 17219 and strfvn 17220). (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 3499 . 2 (𝐹𝑉𝐹 ∈ V)
2 fveq1 6906 . . 3 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
3 df-slot 17216 . . 3 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 fvex 6920 . . 3 (𝐹𝐴) ∈ V
52, 3, 4fvmpt 7016 . 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 1537  wcel 2106  Vcvv 3478  cfv 6563  Slot cslot 17215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-slot 17216
This theorem is referenced by:  bj-evalid  37059  bj-evalidval  37061
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