MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mapprc Structured version   Visualization version   GIF version

Theorem mapprc 7813
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 3933 . . 3 ({𝑓𝑓:𝐴𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 6013 . . . . 5 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 3192 . . . . . 6 𝑓 ∈ V
43dmex 7053 . . . . 5 dom 𝑓 ∈ V
52, 4syl6eqelr 2707 . . . 4 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 1855 . . 3 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 207 . 2 ({𝑓𝑓:𝐴𝐵} ≠ ∅ → 𝐴 ∈ V)
87necon1bi 2818 1 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  Vcvv 3189  c0 3896  dom cdm 5079  wf 5848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-cnv 5087  df-dm 5089  df-rn 5090  df-fn 5855  df-f 5856
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator