Theorem slotslfn 11767

Description: A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.)

Hypothesis

Ref	Expression
slotslfn.e	⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)

Assertion

Ref	Expression
slotslfn	⊢ 𝐸 Fn V

Proof of Theorem slotslfn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2644	. . 3 ⊢ 𝑥 ∈ V
2		slotslfn.e	. . . 4 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
3	2	simpri 112	. . 3 ⊢ (𝐸‘ndx) ∈ ℕ
4	1, 3	fvex 5373	. 2 ⊢ (𝑥‘(𝐸‘ndx)) ∈ V
5	2	simpli 110	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
6		df-slot 11745	. . 3 ⊢ Slot (𝐸‘ndx) = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
7	5, 6	eqtri 2120	. 2 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
8	4, 7	fnmpti 5187	1 ⊢ 𝐸 Fn V

Syntax hints:   ∧ wa 103   = wceq 1299   ∈ wcel 1448  Vcvv 2641   ↦ cmpt 3929   Fn wfn 5054  ‘cfv 5059  ℕcn 8578  ndxcnx 11738  Slot cslot 11740

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067  df-slot 11745

This theorem is referenced by:  slotex  11768  basfn  11798  topontopn  11986