Theorem bj-evalval 36463

Description: Value of the evaluation at a class. (Closed form of strfvnd 17127 and strfvn 17128). (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 3487	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2		fveq1 6884	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓‘𝐴) = (𝐹‘𝐴))
3		df-slot 17124	. . 3 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴))
4		fvex 6898	. . 3 ⊢ (𝐹‘𝐴) ∈ V
5	2, 3, 4	fvmpt 6992	. 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 1533   ∈ wcel 2098  Vcvv 3468  ‘cfv 6537  Slot cslot 17123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-slot 17124

This theorem is referenced by:  bj-evalid  36464  bj-evalidval  36466