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Theorem mapprc 8131
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 4186 . . 3 ({𝑓𝑓:𝐴𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 6290 . . . . 5 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 3417 . . . . . 6 𝑓 ∈ V
43dmex 7366 . . . . 5 dom 𝑓 ∈ V
52, 4syl6eqelr 2915 . . . 4 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 2029 . . 3 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 209 . 2 ({𝑓𝑓:𝐴𝐵} ≠ ∅ → 𝐴 ∈ V)
87necon1bi 3027 1 𝐴 ∈ V → {𝑓𝑓:𝐴𝐵} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1656  wex 1878  wcel 2164  {cab 2811  wne 2999  Vcvv 3414  c0 4146  dom cdm 5346  wf 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-cnv 5354  df-dm 5356  df-rn 5357  df-fn 6130  df-f 6131
This theorem is referenced by: (None)
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