Theorem strslfv2 12662

Description: A variation on strslfv 12663 to avoid asserting that S itself is a function, which involves sethood of all the ordered pair components of S. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strfv2.s	|- S e. _V
strfv2.f	|- Fun `' `' S
strslfv2.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strfv2.n	|- <. ( E ` ndx ) , C >. e. S

Assertion

Ref	Expression
strslfv2	|- ( C e. V -> C = ( E ` S ) )

Proof of Theorem strslfv2

Step	Hyp	Ref	Expression
1		strslfv2.e	. 2 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2		strfv2.s	. . 3 |- S e. _V
3	2	a1i 9	. 2 |- ( C e. V -> S e. _V )
4		strfv2.f	. . 3 |- Fun `' `' S
5	4	a1i 9	. 2 |- ( C e. V -> Fun `' `' S )
6		strfv2.n	. . 3 |- <. ( E ` ndx ) , C >. e. S
7	6	a1i 9	. 2 |- ( C e. V -> <. ( E ` ndx ) , C >. e. S )
8		id 19	. 2 |- ( C e. V -> C e. V )
9	1, 3, 5, 7, 8	strslfv2d 12661	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   <.cop 3621   `'ccnv 4658   Fun wfun 5248   ` cfv 5254   NNcn 8982   ndxcnx 12615  Slot cslot 12617

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-slot 12622

This theorem is referenced by: (None)