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Theorem fsetprcnex 8902
Description: The class of all functions from a nonempty set 𝐴 into a proper class 𝐵 is not a set. If one of the preconditions is not fufilled, then {𝑓𝑓:𝐴𝐵} is a set, see fsetdmprc0 8895 for 𝐴 ∉ V, fset0 8894 for 𝐴 = ∅, and fsetex 8896 for 𝐵 ∈ V, see also fsetexb 8904. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.)
Assertion
Ref Expression
fsetprcnex (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem fsetprcnex
Dummy variables 𝑎 𝑔 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4353 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
2 feq1 6716 . . . . . . . . . 10 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
32cbvabv 2812 . . . . . . . . 9 {𝑓𝑓:𝐴𝐵} = {𝑚𝑚:𝐴𝐵}
4 fveq1 6905 . . . . . . . . . 10 (𝑔 = 𝑛 → (𝑔𝑎) = (𝑛𝑎))
54cbvmptv 5255 . . . . . . . . 9 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)) = (𝑛 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑛𝑎))
63, 5fsetfocdm 8901 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵)
7 focdmex 7980 . . . . . . . 8 ({𝑓𝑓:𝐴𝐵} ∈ V → ((𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵𝐵 ∈ V))
86, 7syl5com 31 . . . . . . 7 ((𝐴𝑉𝑎𝐴) → ({𝑓𝑓:𝐴𝐵} ∈ V → 𝐵 ∈ V))
98nelcon3d 3050 . . . . . 6 ((𝐴𝑉𝑎𝐴) → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V))
109expcom 413 . . . . 5 (𝑎𝐴 → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
1110exlimiv 1930 . . . 4 (∃𝑎 𝑎𝐴 → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
121, 11sylbi 217 . . 3 (𝐴 ≠ ∅ → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
1312impcom 407 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V))
1413imp 406 1 (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1779  wcel 2108  {cab 2714  wne 2940  wnel 3046  Vcvv 3480  c0 4333  cmpt 5225  wf 6557  ontowfo 6559  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  fsetcdmex  8903  fsetexb  8904
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