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Theorem mapprc 8869
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 4391 . . 3 ({𝑓𝑓:𝐴𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 6746 . . . . 5 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 3482 . . . . . 6 𝑓 ∈ V
43dmex 7932 . . . . 5 dom 𝑓 ∈ V
52, 4eqeltrrdi 2848 . . . 4 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 1928 . . 3 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 217 . 2 ({𝑓𝑓:𝐴𝐵} ≠ ∅ → 𝐴 ∈ V)
87necon1bi 2967 1 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  Vcvv 3478  c0 4339  dom cdm 5689  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-fn 6566  df-f 6567
This theorem is referenced by:  efmndbasabf  18898
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