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Theorem fvrn0 6922
Description: A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)
Assertion
Ref Expression
fvrn0 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})

Proof of Theorem fvrn0
StepHypRef Expression
1 id 22 . . 3 ((𝐹𝑋) = ∅ → (𝐹𝑋) = ∅)
2 ssun2 4174 . . . 4 {∅} ⊆ (ran 𝐹 ∪ {∅})
3 0ex 5308 . . . . 5 ∅ ∈ V
43snid 4665 . . . 4 ∅ ∈ {∅}
52, 4sselii 3980 . . 3 ∅ ∈ (ran 𝐹 ∪ {∅})
61, 5eqeltrdi 2842 . 2 ((𝐹𝑋) = ∅ → (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}))
7 ssun1 4173 . . 3 ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8 fvprc 6884 . . . . 5 𝑋 ∈ V → (𝐹𝑋) = ∅)
98con1i 147 . . . 4 (¬ (𝐹𝑋) = ∅ → 𝑋 ∈ V)
10 fvexd 6907 . . . 4 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ V)
11 fvbr0 6921 . . . . . 6 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
1211ori 860 . . . . 5 𝑋𝐹(𝐹𝑋) → (𝐹𝑋) = ∅)
1312con1i 147 . . . 4 (¬ (𝐹𝑋) = ∅ → 𝑋𝐹(𝐹𝑋))
14 brelrng 5941 . . . 4 ((𝑋 ∈ V ∧ (𝐹𝑋) ∈ V ∧ 𝑋𝐹(𝐹𝑋)) → (𝐹𝑋) ∈ ran 𝐹)
159, 10, 13, 14syl3anc 1372 . . 3 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ ran 𝐹)
167, 15sselid 3981 . 2 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}))
176, 16pm2.61i 182 1 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  cun 3947  c0 4323  {csn 4629   class class class wbr 5149  ran crn 5678  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688  df-iota 6496  df-fv 6552
This theorem is referenced by:  fvn0fvelrn  6923  fvssunirnOLD  6926  orderseqlem  8143  dfac4  10117  dfac2b  10125  dfacacn  10136  axdc2lem  10443  axcclem  10452  seqexw  13982  plusffval  18567  grpsubfval  18868  mulgfval  18952  staffval  20455  scaffval  20490  lpival  20883  ipffval  21201  nmfval  24097  tcphex  24734  tchnmfval  24745  rrnval  36695  lsatset  37860  fvnonrel  42348
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