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Theorem strfvd 13461
Description: Deduction version of strfv 13464. (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 13447 . 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 5733 . . 3  |-  ( Fun 
S  ->  ( <. ( E `  ndx ) ,  C >.  e.  S  ->  ( S `  ( E `  ndx ) )  =  C ) )
74, 5, 6sylc 58 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
83, 7eqtr2d 2445 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   <.cop 3785   Fun wfun 5415   ` cfv 5421   ndxcnx 13429  Slot cslot 13431
This theorem is referenced by:  strssd  13466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-slot 13436
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