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Theorem mapprc 6586
 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
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 abn0m 3415 . . . 4 (∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 5318 . . . . . 6 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 2712 . . . . . . 7 𝑓 ∈ V
43dmex 4845 . . . . . 6 dom 𝑓 ∈ V
52, 4eqeltrrdi 2246 . . . . 5 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 1575 . . . 4 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 120 . . 3 (∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} → 𝐴 ∈ V)
87con3i 622 . 2 𝐴 ∈ V → ¬ ∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵})
9 notm0 3410 . 2 (¬ ∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ {𝑓𝑓:𝐴𝐵} = ∅)
108, 9sylib 121 1 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332  ∃wex 1469   ∈ wcel 2125  {cab 2140  Vcvv 2709  ∅c0 3390  dom cdm 4579  ⟶wf 5159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-cnv 4587  df-dm 4589  df-rn 4590  df-fn 5166  df-f 5167 This theorem is referenced by: (None)
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