Theorem strfv2 15906

Description: A variation on strfv 15907 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.)

Hypotheses

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

Assertion

Ref	Expression
strfv2	|- ( C e. V -> C = ( E ` S ) )

Proof of Theorem strfv2

Step	Hyp	Ref	Expression
1		strfv2.e	. 2 |- E = Slot ( E ` ndx )
2		strfv2.s	. . 3 |- S e. _V
3	2	a1i 11	. 2 |- ( C e. V -> S e. _V )
4		strfv2.f	. . 3 |- Fun `' `' S
5	4	a1i 11	. 2 |- ( C e. V -> Fun `' `' S )
6		strfv2.n	. . 3 |- <. ( E ` ndx ) , C >. e. S
7	6	a1i 11	. 2 |- ( C e. V -> <. ( E ` ndx ) , C >. e. S )
8		id 22	. 2 |- ( C e. V -> C e. V )
9	1, 3, 5, 7, 8	strfv2d 15905	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   `'ccnv 5113   Fun wfun 5882   ` cfv 5888   ndxcnx 15854  Slot cslot 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861

This theorem is referenced by:  strfv  15907