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Theorem bj-evalval 33152
Description: Value of the evaluation at a class. (Closed form of strfvnd 15923 and strfvn 15926). (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 3243 . 2 (𝐹𝑉𝐹 ∈ V)
2 fveq1 6228 . . 3 (𝑓 = 𝐹 → (𝑓𝐴) = (𝐹𝐴))
3 df-slot 15908 . . 3 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 fvex 6239 . . 3 (𝐹𝐴) ∈ V
52, 3, 4fvmpt 6321 . 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 1523  wcel 2030  Vcvv 3231  cfv 5926  Slot cslot 15903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-slot 15908
This theorem is referenced by:  bj-evalid  33153  bj-evalidval  33156
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