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Theorem elbasfv 15901
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 = (𝐹𝑍)
elbasfv.b 𝐵 = (Base‘𝑆)
Assertion
Ref Expression
elbasfv (𝑋𝐵𝑍 ∈ V)

Proof of Theorem elbasfv
StepHypRef Expression
1 n0i 3912 . 2 (𝑋𝐵 → ¬ 𝐵 = ∅)
2 elbasfv.s . . . . 5 𝑆 = (𝐹𝑍)
3 fvprc 6172 . . . . 5 𝑍 ∈ V → (𝐹𝑍) = ∅)
42, 3syl5eq 2666 . . . 4 𝑍 ∈ V → 𝑆 = ∅)
54fveq2d 6182 . . 3 𝑍 ∈ V → (Base‘𝑆) = (Base‘∅))
6 elbasfv.b . . 3 𝐵 = (Base‘𝑆)
7 base0 15893 . . 3 ∅ = (Base‘∅)
85, 6, 73eqtr4g 2679 . 2 𝑍 ∈ V → 𝐵 = ∅)
91, 8nsyl2 142 1 (𝑋𝐵𝑍 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  cfv 5876  Basecbs 15838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-slot 15842  df-base 15844
This theorem is referenced by:  frmdelbas  17371  symginv  17803  symggen  17871  psgneu  17907  psgnpmtr  17911  coe1sfi  19564  frgpcyg  19903  lindfind  20136  q1pval  23894  r1pval  23897
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