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Theorem elbasfv 13207
Description: Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Hypotheses
Ref Expression
elbasfv.s  |-  S  =  ( F `  Z
)
elbasfv.b  |-  B  =  ( Base `  S
)
Assertion
Ref Expression
elbasfv  |-  ( X  e.  B  ->  Z  e.  _V )

Proof of Theorem elbasfv
StepHypRef Expression
1 n0i 3473 . 2  |-  ( X  e.  B  ->  -.  B  =  (/) )
2 elbasfv.s . . . . 5  |-  S  =  ( F `  Z
)
3 fvprc 5535 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F `  Z )  =  (/) )
42, 3syl5eq 2340 . . . 4  |-  ( -.  Z  e.  _V  ->  S  =  (/) )
54fveq2d 5545 . . 3  |-  ( -.  Z  e.  _V  ->  (
Base `  S )  =  ( Base `  (/) ) )
6 elbasfv.b . . 3  |-  B  =  ( Base `  S
)
7 base0 13201 . . 3  |-  (/)  =  (
Base `  (/) )
85, 6, 73eqtr4g 2353 . 2  |-  ( -.  Z  e.  _V  ->  B  =  (/) )
91, 8nsyl2 119 1  |-  ( X  e.  B  ->  Z  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ` cfv 5271   Basecbs 13164
This theorem is referenced by:  frmdelbas  14491  symginv  14798  coe1sfi  16309  frgpcyg  16543  q1pval  19555  r1pval  19558  lindfind  27389  symggen  27514  psgneu  27532  psgnpmtr  27536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168  df-base 13169
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