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Theorem elbasfv 16533
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 4280 . 2 (𝑋𝐵 → ¬ 𝐵 = ∅)
2 elbasfv.s . . . . 5 𝑆 = (𝐹𝑍)
3 fvprc 6644 . . . . 5 𝑍 ∈ V → (𝐹𝑍) = ∅)
42, 3syl5eq 2871 . . . 4 𝑍 ∈ V → 𝑆 = ∅)
54fveq2d 6655 . . 3 𝑍 ∈ V → (Base‘𝑆) = (Base‘∅))
6 elbasfv.b . . 3 𝐵 = (Base‘𝑆)
7 base0 16525 . . 3 ∅ = (Base‘∅)
85, 6, 73eqtr4g 2884 . 2 𝑍 ∈ V → 𝐵 = ∅)
91, 8nsyl2 143 1 (𝑋𝐵𝑍 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  Vcvv 3479  c0 4274  cfv 6336  Basecbs 16472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-slot 16476  df-base 16478
This theorem is referenced by:  frmdelbas  18007  symginv  18519  symggen  18587  psgneu  18623  psgnpmtr  18627  coe1sfi  20367  frgpcyg  20706  lindfind  20946  q1pval  24743  r1pval  24746  symgsubg  30749
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