MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  strssd Structured version   Unicode version

Theorem strssd 13495
Description: Deduction version of strss 13496. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
strssd.e  |-  E  = Slot  ( E `  ndx )
strssd.t  |-  ( ph  ->  T  e.  V )
strssd.f  |-  ( ph  ->  Fun  T )
strssd.s  |-  ( ph  ->  S  C_  T )
strssd.n  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
Assertion
Ref Expression
strssd  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )

Proof of Theorem strssd
StepHypRef Expression
1 strssd.e . . 3  |-  E  = Slot  ( E `  ndx )
2 strssd.t . . 3  |-  ( ph  ->  T  e.  V )
3 strssd.f . . 3  |-  ( ph  ->  Fun  T )
4 strssd.s . . . 4  |-  ( ph  ->  S  C_  T )
5 strssd.n . . . 4  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
64, 5sseldd 3341 . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  T )
71, 2, 3, 6strfvd 13490 . 2  |-  ( ph  ->  C  =  ( E `
 T ) )
82, 4ssexd 4342 . . 3  |-  ( ph  ->  S  e.  _V )
9 funss 5464 . . . 4  |-  ( S 
C_  T  ->  ( Fun  T  ->  Fun  S ) )
104, 3, 9sylc 58 . . 3  |-  ( ph  ->  Fun  S )
111, 8, 10, 5strfvd 13490 . 2  |-  ( ph  ->  C  =  ( E `
 S ) )
127, 11eqtr3d 2469 1  |-  ( ph  ->  ( E `  T
)  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   <.cop 3809   Fun wfun 5440   ` cfv 5446   ndxcnx 13458  Slot cslot 13460
This theorem is referenced by:  strss  13496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-slot 13465
  Copyright terms: Public domain W3C validator