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Theorem mapprc 8772
Description: When 𝐴 is a proper class, the class of all functions mapping 𝐴 to 𝐵 is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
mapprc 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapprc
StepHypRef Expression
1 abn0 4341 . . 3 ({𝑓𝑓:𝐴𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 6678 . . . . 5 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 3448 . . . . . 6 𝑓 ∈ V
43dmex 7849 . . . . 5 dom 𝑓 ∈ V
52, 4eqeltrrdi 2843 . . . 4 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 1934 . . 3 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 216 . 2 ({𝑓𝑓:𝐴𝐵} ≠ ∅ → 𝐴 ∈ V)
87necon1bi 2969 1 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2940  Vcvv 3444  c0 4283  dom cdm 5634  wf 6493
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 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645  df-fn 6500  df-f 6501
This theorem is referenced by:  efmndbasabf  18687
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