Theorem strfvd 15904

Description: Deduction version of strfv 15907. (Contributed by Mario Carneiro, 15-Nov-2014.)

Hypotheses

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

Assertion

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

Proof of Theorem strfvd

Step	Hyp	Ref	Expression
1		strfvd.e	. . 3 |- E = Slot ( E ` ndx )
2		strfvd.s	. . 3 |- ( ph -> S e. V )
3	1, 2	strfvnd 15876	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
4		strfvd.f	. . 3 |- ( ph -> Fun S )
5		strfvd.n	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
6		funopfv 6235	. . 3 |- ( Fun S -> ( <. ( E ` ndx ) , C >. e. S -> ( S ` ( E ` ndx ) ) = C ) )
7	4, 5, 6	sylc 65	. 2 |- ( ph -> ( S ` ( E ` ndx ) ) = C )
8	3, 7	eqtr2d 2657	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   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-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861

This theorem is referenced by:  strssd  15909