Theorem slotslfn 13356

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 2824	. . 3 ⊢ 𝑥 ∈ V
2		slotslfn.e	. . . 4 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
3	2	simpri 113	. . 3 ⊢ (𝐸‘ndx) ∈ ℕ
4	1, 3	fvex 5710	. 2 ⊢ (𝑥‘(𝐸‘ndx)) ∈ V
5	2	simpli 111	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
6		df-slot 13334	. . 3 ⊢ Slot (𝐸‘ndx) = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
7	5, 6	eqtri 2259	. 2 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
8	4, 7	fnmpti 5507	1 ⊢ 𝐸 Fn V

Syntax hints:   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4187   Fn wfn 5367  ‘cfv 5372  ℕcn 9283  ndxcnx 13327  Slot cslot 13329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-slot 13334

This theorem is referenced by:  slotex  13357  basfn  13389  topontopn  15061