Theorem bj-evalval 37774

Description: Value of the evaluation at a class. Closed form of strfvnd 17267 and strfvn 17268. (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 3478	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		fveq1 6884	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓‘𝐴) = (𝐹‘𝐴))
3		df-slot 17264	. . 3 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴))
4		fvex 6898	. . 3 ⊢ (𝐹‘𝐴) ∈ V
5	2, 3, 4	fvmpt 6993	. 2 ⊢ (𝐹 ∈ V → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))
6	1, 5	syl 18	1 ⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ‘cfv 6540  Slot cslot 17263

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-slot 17264

This theorem is used by:  bj-evalid  37775  bj-evalidval  37777