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Theorem mapprc 6512
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 3356 . . . 4 (∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ ∃𝑓 𝑓:𝐴𝐵)
2 fdm 5246 . . . . . 6 (𝑓:𝐴𝐵 → dom 𝑓 = 𝐴)
3 vex 2661 . . . . . . 7 𝑓 ∈ V
43dmex 4773 . . . . . 6 dom 𝑓 ∈ V
52, 4syl6eqelr 2207 . . . . 5 (𝑓:𝐴𝐵𝐴 ∈ V)
65exlimiv 1560 . . . 4 (∃𝑓 𝑓:𝐴𝐵𝐴 ∈ V)
71, 6sylbi 120 . . 3 (∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵} → 𝐴 ∈ V)
87con3i 604 . 2 𝐴 ∈ V → ¬ ∃𝑔 𝑔 ∈ {𝑓𝑓:𝐴𝐵})
9 notm0 3351 . 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 1314  wex 1451  wcel 1463  {cab 2101  Vcvv 2658  c0 3331  dom cdm 4507  wf 5087
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518  df-fn 5094  df-f 5095
This theorem is referenced by: (None)
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