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Theorem strfvd 14326
Description: Deduction version of strfv 14329. (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
StepHypRef Expression
1 strfvd.e . . 3  |-  E  = Slot  ( E `  ndx )
2 strfvd.s . . 3  |-  ( ph  ->  S  e.  V )
31, 2strfvnd 14310 . 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 5843 . . 3  |-  ( Fun 
S  ->  ( <. ( E `  ndx ) ,  C >.  e.  S  ->  ( S `  ( E `  ndx ) )  =  C ) )
74, 5, 6sylc 60 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
83, 7eqtr2d 2496 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   <.cop 3994   Fun wfun 5523   ` cfv 5529   ndxcnx 14292  Slot cslot 14294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-slot 14299
This theorem is referenced by:  strssd  14331
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