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Theorem fvrn0 6558
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 4065 . . . 4 {∅} ⊆ (ran 𝐹 ∪ {∅})
3 0ex 5096 . . . . 5 ∅ ∈ V
43snid 4500 . . . 4 ∅ ∈ {∅}
52, 4sselii 3881 . . 3 ∅ ∈ (ran 𝐹 ∪ {∅})
61, 5syl6eqel 2889 . 2 ((𝐹𝑋) = ∅ → (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}))
7 ssun1 4064 . . 3 ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8 fvprc 6523 . . . . 5 𝑋 ∈ V → (𝐹𝑋) = ∅)
98con1i 149 . . . 4 (¬ (𝐹𝑋) = ∅ → 𝑋 ∈ V)
10 fvexd 6545 . . . 4 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ V)
11 fvbr0 6557 . . . . . 6 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
1211ori 856 . . . . 5 𝑋𝐹(𝐹𝑋) → (𝐹𝑋) = ∅)
1312con1i 149 . . . 4 (¬ (𝐹𝑋) = ∅ → 𝑋𝐹(𝐹𝑋))
14 brelrng 5685 . . . 4 ((𝑋 ∈ V ∧ (𝐹𝑋) ∈ V ∧ 𝑋𝐹(𝐹𝑋)) → (𝐹𝑋) ∈ ran 𝐹)
159, 10, 13, 14syl3anc 1362 . . 3 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ ran 𝐹)
167, 15sseldi 3882 . 2 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}))
176, 16pm2.61i 183 1 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1520  wcel 2079  Vcvv 3432  cun 3852  c0 4206  {csn 4466   class class class wbr 4956  ran crn 5436  cfv 6217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-cnv 5443  df-dm 5445  df-rn 5446  df-iota 6181  df-fv 6225
This theorem is referenced by:  fvssunirn  6559  dfac4  9383  dfac2b  9391  dfacacn  9402  axdc2lem  9705  axcclem  9714  seqexw  13223  plusffval  17674  grpsubfval  17892  mulgfval  17971  staffval  19296  scaffval  19330  lpival  19695  ipffval  20462  nmfval  22869  tcphex  23491  tchnmfval  23502  orderseqlem  32648  rrnval  34583  lsatset  35607  fvnonrel  39393
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