Theorem bj-evalval 37041

Description: Value of the evaluation at a class. (Closed form of strfvnd 17232 and strfvn 17233). (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.)

Assertion

Ref	Expression
bj-evalval	⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))

Proof of Theorem bj-evalval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3509	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		fveq1 6919	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓‘𝐴) = (𝐹‘𝐴))
3		df-slot 17229	. . 3 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴))
4		fvex 6933	. . 3 ⊢ (𝐹‘𝐴) ∈ V
5	2, 3, 4	fvmpt 7029	. 2 ⊢ (𝐹 ∈ V → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))
6	1, 5	syl 17	1 ⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ‘cfv 6573  Slot cslot 17228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-slot 17229

This theorem is referenced by:  bj-evalid  37042  bj-evalidval  37044