Theorem slotslfn 13058

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 2802	. . 3 |- x e. _V
2		slotslfn.e	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3	2	simpri 113	. . 3 |- ( E ` ndx ) e. NN
4	1, 3	fvex 5647	. 2 |- ( x ` ( E ` ndx ) ) e. _V
5	2	simpli 111	. . 3 |- E = Slot ( E ` ndx )
6		df-slot 13036	. . 3 |- Slot ( E ` ndx ) = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
7	5, 6	eqtri 2250	. 2 |- E = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
8	4, 7	fnmpti 5452	1 |- E Fn _V

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   |-> cmpt 4145   Fn wfn 5313   ` cfv 5318   NNcn 9110   ndxcnx 13029  Slot cslot 13031

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-slot 13036

This theorem is referenced by:  slotex  13059  basfn  13091  topontopn  14711