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Theorem elbasfv 13505
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 3626 . 2  |-  ( X  e.  B  ->  -.  B  =  (/) )
2 elbasfv.s . . . . 5  |-  S  =  ( F `  Z
)
3 fvprc 5715 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F `  Z )  =  (/) )
42, 3syl5eq 2480 . . . 4  |-  ( -.  Z  e.  _V  ->  S  =  (/) )
54fveq2d 5725 . . 3  |-  ( -.  Z  e.  _V  ->  (
Base `  S )  =  ( Base `  (/) ) )
6 elbasfv.b . . 3  |-  B  =  ( Base `  S
)
7 base0 13499 . . 3  |-  (/)  =  (
Base `  (/) )
85, 6, 73eqtr4g 2493 . 2  |-  ( -.  Z  e.  _V  ->  B  =  (/) )
91, 8nsyl2 121 1  |-  ( X  e.  B  ->  Z  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2949   (/)c0 3621   ` cfv 5447   Basecbs 13462
This theorem is referenced by:  frmdelbas  14791  symginv  15098  coe1sfi  16603  frgpcyg  16847  q1pval  20069  r1pval  20072  lindfind  27255  symggen  27380  psgneu  27398  psgnpmtr  27402
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-slot 13466  df-base 13467
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