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Theorem strfv2 13502
Description: A variation on strfv 13503 to avoid asserting that  S itself is a function, which involves sethood of all the ordered pair components of  S. (Contributed by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
strfv2.s  |-  S  e. 
_V
strfv2.f  |-  Fun  `' `' S
strfv2.e  |-  E  = Slot  ( E `  ndx )
strfv2.n  |-  <. ( E `  ndx ) ,  C >.  e.  S
Assertion
Ref Expression
strfv2  |-  ( C  e.  V  ->  C  =  ( E `  S ) )

Proof of Theorem strfv2
StepHypRef Expression
1 strfv2.e . 2  |-  E  = Slot  ( E `  ndx )
2 strfv2.s . . 3  |-  S  e. 
_V
32a1i 11 . 2  |-  ( C  e.  V  ->  S  e.  _V )
4 strfv2.f . . 3  |-  Fun  `' `' S
54a1i 11 . 2  |-  ( C  e.  V  ->  Fun  `' `' S )
6 strfv2.n . . 3  |-  <. ( E `  ndx ) ,  C >.  e.  S
76a1i 11 . 2  |-  ( C  e.  V  ->  <. ( E `  ndx ) ,  C >.  e.  S
)
8 id 21 . 2  |-  ( C  e.  V  ->  C  e.  V )
91, 3, 5, 7, 8strfv2d 13501 1  |-  ( C  e.  V  ->  C  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   `'ccnv 4879   Fun wfun 5450   ` cfv 5456   ndxcnx 13468  Slot cslot 13470
This theorem is referenced by:  strfv  13503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pr 4405
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464  df-slot 13475
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