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Theorem strfvd 13503
Description: Deduction version of strfv 13506. (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 13489 . 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 5769 . . 3  |-  ( Fun 
S  ->  ( <. ( E `  ndx ) ,  C >.  e.  S  ->  ( S `  ( E `  ndx ) )  =  C ) )
74, 5, 6sylc 59 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
83, 7eqtr2d 2471 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   <.cop 3819   Fun wfun 5451   ` cfv 5457   ndxcnx 13471  Slot cslot 13473
This theorem is referenced by:  strssd  13508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-slot 13478
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