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Theorem mapprc 8402
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 4319 . . 3 ({𝑓𝑓:𝐴𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 6511 . . . . 5 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 3483 . . . . . 6 𝑓 ∈ V
43dmex 7608 . . . . 5 dom 𝑓 ∈ V
52, 4eqeltrrdi 2925 . . . 4 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 1932 . . 3 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 220 . 2 ({𝑓𝑓:𝐴𝐵} ≠ ∅ → 𝐴 ∈ V)
87necon1bi 3042 1 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wne 3014  Vcvv 3480  c0 4276  dom cdm 5543  wf 6340
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 5190  ax-nul 5197  ax-pr 5318  ax-un 7452
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-ne 3015  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-cnv 5551  df-dm 5553  df-rn 5554  df-fn 6347  df-f 6348
This theorem is referenced by:  efmndbasabf  18035
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