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Theorem fvrn0 6439
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 3983 . . . 4 {∅} ⊆ (ran 𝐹 ∪ {∅})
3 0ex 4991 . . . . 5 ∅ ∈ V
43snid 4409 . . . 4 ∅ ∈ {∅}
52, 4sselii 3802 . . 3 ∅ ∈ (ran 𝐹 ∪ {∅})
61, 5syl6eqel 2900 . 2 ((𝐹𝑋) = ∅ → (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}))
7 ssun1 3982 . . 3 ran 𝐹 ⊆ (ran 𝐹 ∪ {∅})
8 fvprc 6404 . . . . 5 𝑋 ∈ V → (𝐹𝑋) = ∅)
98con1i 146 . . . 4 (¬ (𝐹𝑋) = ∅ → 𝑋 ∈ V)
10 fvexd 6426 . . . 4 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ V)
11 fvbr0 6438 . . . . . 6 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
1211ori 879 . . . . 5 𝑋𝐹(𝐹𝑋) → (𝐹𝑋) = ∅)
1312con1i 146 . . . 4 (¬ (𝐹𝑋) = ∅ → 𝑋𝐹(𝐹𝑋))
14 brelrng 5563 . . . 4 ((𝑋 ∈ V ∧ (𝐹𝑋) ∈ V ∧ 𝑋𝐹(𝐹𝑋)) → (𝐹𝑋) ∈ ran 𝐹)
159, 10, 13, 14syl3anc 1483 . . 3 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ ran 𝐹)
167, 15sseldi 3803 . 2 (¬ (𝐹𝑋) = ∅ → (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}))
176, 16pm2.61i 176 1 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1637  wcel 2157  Vcvv 3398  cun 3774  c0 4123  {csn 4377   class class class wbr 4851  ran crn 5319  cfv 6104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-cnv 5326  df-dm 5328  df-rn 5329  df-iota 6067  df-fv 6112
This theorem is referenced by:  fvssunirn  6440  dfac4  9231  dfac2b  9239  dfac2OLD  9241  dfacacn  9251  axdc2lem  9558  axcclem  9567  plusffval  17455  staffval  19054  scaffval  19088  lpival  19457  ipffval  20206  nmfval  22610  tchex  23232  tchnmfval  23243  orderseqlem  32078  rrnval  33939  lsatset  34772  fvnonrel  38404
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