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Theorem elbasfv 14528
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 3785 . 2  |-  ( X  e.  B  ->  -.  B  =  (/) )
2 elbasfv.s . . . . 5  |-  S  =  ( F `  Z
)
3 fvprc 5853 . . . . 5  |-  ( -.  Z  e.  _V  ->  ( F `  Z )  =  (/) )
42, 3syl5eq 2515 . . . 4  |-  ( -.  Z  e.  _V  ->  S  =  (/) )
54fveq2d 5863 . . 3  |-  ( -.  Z  e.  _V  ->  (
Base `  S )  =  ( Base `  (/) ) )
6 elbasfv.b . . 3  |-  B  =  ( Base `  S
)
7 base0 14520 . . 3  |-  (/)  =  (
Base `  (/) )
85, 6, 73eqtr4g 2528 . 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 1374    e. wcel 1762   _Vcvv 3108   (/)c0 3780   ` cfv 5581   Basecbs 14481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-slot 14485  df-base 14486
This theorem is referenced by:  frmdelbas  15839  symginv  16217  symggen  16286  psgneu  16322  psgnpmtr  16326  coe1sfi  18018  coe1sfiOLD  18019  frgpcyg  18374  lindfind  18613  q1pval  22284  r1pval  22287
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