Theorem strfv2d 15905

Description: Deduction version of strfv 15907. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
strfv2d.e	|- E = Slot ( E ` ndx )
strfv2d.s	|- ( ph -> S e. V )
strfv2d.f	|- ( ph -> Fun `' `' S )
strfv2d.n	|- ( ph -> <. ( E ` ndx ) , C >. e. S )
strfv2d.c	|- ( ph -> C e. W )

Assertion

Ref	Expression
strfv2d	|- ( ph -> C = ( E ` S ) )

Proof of Theorem strfv2d

Step	Hyp	Ref	Expression
1		strfv2d.e	. . 3 |- E = Slot ( E ` ndx )
2		strfv2d.s	. . 3 |- ( ph -> S e. V )
3	1, 2	strfvnd 15876	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
4		cnvcnv2 5588	. . . . 5 |- `' `' S = ( S |` _V )
5	4	fveq1i 6192	. . . 4 |- ( `' `' S ` ( E ` ndx ) ) = ( ( S |` _V ) ` ( E ` ndx ) )
6		fvex 6201	. . . . 5 |- ( E ` ndx ) e. _V
7		fvres 6207	. . . . 5 |- ( ( E ` ndx ) e. _V -> ( ( S |` _V ) ` ( E ` ndx ) ) = ( S ` ( E ` ndx ) ) )
8	6, 7	ax-mp 5	. . . 4 |- ( ( S |` _V ) ` ( E ` ndx ) ) = ( S ` ( E ` ndx ) )
9	5, 8	eqtri 2644	. . 3 |- ( `' `' S ` ( E ` ndx ) ) = ( S ` ( E ` ndx ) )
10		strfv2d.f	. . . 4 |- ( ph -> Fun `' `' S )
11		strfv2d.n	. . . . . 6 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
12		strfv2d.c	. . . . . . . 8 |- ( ph -> C e. W )
13		elex 3212	. . . . . . . 8 |- ( C e. W -> C e. _V )
14	12, 13	syl 17	. . . . . . 7 |- ( ph -> C e. _V )
15		opelxpi 5148	. . . . . . 7 |- ( ( ( E ` ndx ) e. _V /\ C e. _V ) -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
16	6, 14, 15	sylancr 695	. . . . . 6 |- ( ph -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
17	11, 16	elind 3798	. . . . 5 |- ( ph -> <. ( E ` ndx ) , C >. e. ( S i^i ( _V X. _V ) ) )
18		cnvcnv 5586	. . . . 5 |- `' `' S = ( S i^i ( _V X. _V ) )
19	17, 18	syl6eleqr 2712	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. `' `' S )
20		funopfv 6235	. . . 4 |- ( Fun `' `' S -> ( <. ( E ` ndx ) , C >. e. `' `' S -> ( `' `' S ` ( E ` ndx ) ) = C ) )
21	10, 19, 20	sylc 65	. . 3 |- ( ph -> ( `' `' S ` ( E ` ndx ) ) = C )
22	9, 21	syl5eqr 2670	. 2 |- ( ph -> ( S ` ( E ` ndx ) ) = C )
23	3, 22	eqtr2d 2657	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   <.cop 4183   X. cxp 5112   `'ccnv 5113   |` cres 5116   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:  strfv2  15906  opelstrbas  15978  eengbas  25861  ebtwntg  25862  ecgrtg  25863  elntg  25864  edgfiedgval  25902