Theorem strslfv2 12459

Description: A variation on strslfv 12460 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 12458	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   <.cop 3586   `'ccnv 4610   Fun wfun 5192   ` 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-sbc 2956  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-res 4623  df-iota 5160  df-fun 5200  df-fv 5206  df-slot 12420

This theorem is referenced by: (None)