Theorem slotslfn 12442

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 ) e. NN )

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 2733	. . 3 |- x e. _V
2		slotslfn.e	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3	2	simpri 112	. . 3 |- ( E ` ndx ) e. NN
4	1, 3	fvex 5516	. 2 |- ( x ` ( E ` ndx ) ) e. _V
5	2	simpli 110	. . 3 |- E = Slot ( E ` ndx )
6		df-slot 12420	. . 3 |- Slot ( E ` ndx ) = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
7	5, 6	eqtri 2191	. 2 |- E = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
8	4, 7	fnmpti 5326	1 |- E Fn _V

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   |-> cmpt 4050   Fn wfn 5193   ` cfv 5198   NNcn 8878   ndxcnx 12413  Slot cslot 12415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-slot 12420

This theorem is referenced by:  slotex  12443  basfn  12473  topontopn  12829