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Theorem elbasfv 14220
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 3641 . 2  |-  ( X  e.  B  ->  -.  B  =  (/) )
2 elbasfv.s . . . . 5  |-  S  =  ( F `  Z
)
3 fvprc 5684 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F `  Z )  =  (/) )
42, 3syl5eq 2486 . . . 4  |-  ( -.  Z  e.  _V  ->  S  =  (/) )
54fveq2d 5694 . . 3  |-  ( -.  Z  e.  _V  ->  (
Base `  S )  =  ( Base `  (/) ) )
6 elbasfv.b . . 3  |-  B  =  ( Base `  S
)
7 base0 14212 . . 3  |-  (/)  =  (
Base `  (/) )
85, 6, 73eqtr4g 2499 . 2  |-  ( -.  Z  e.  _V  ->  B  =  (/) )
91, 8nsyl2 127 1  |-  ( X  e.  B  ->  Z  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2971   (/)c0 3636   ` cfv 5417   Basecbs 14173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-slot 14177  df-base 14178
This theorem is referenced by:  frmdelbas  15530  symginv  15906  symggen  15975  psgneu  16011  psgnpmtr  16015  coe1sfi  17667  coe1sfiOLD  17668  frgpcyg  18005  lindfind  18244  q1pval  21624  r1pval  21627
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