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Theorem mapprc 6409
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 3308 . . . 4 (∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 5166 . . . . . 6 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 2622 . . . . . . 7 𝑓 ∈ V
43dmex 4699 . . . . . 6 dom 𝑓 ∈ V
52, 4syl6eqelr 2179 . . . . 5 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 1534 . . . 4 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 119 . . 3 (∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} → 𝐴 ∈ V)
87con3i 597 . 2 𝐴 ∈ V → ¬ ∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵})
9 notm0 3303 . 2 (¬ ∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ {𝑓𝑓:𝐴𝐵} = ∅)
108, 9sylib 120 1 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wex 1426  wcel 1438  {cab 2074  Vcvv 2619  c0 3286  dom cdm 4438  wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449  df-fn 5018  df-f 5019
This theorem is referenced by: (None)
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