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Theorem fsetprcnex 8901
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 8894 for 𝐴 ∉ V, fset0 8893 for 𝐴 = ∅, and fsetex 8895 for 𝐵 ∈ V, see also fsetexb 8903. (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 4359 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
2 feq1 6717 . . . . . . . . . 10 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
32cbvabv 2810 . . . . . . . . 9 {𝑓𝑓:𝐴𝐵} = {𝑚𝑚:𝐴𝐵}
4 fveq1 6906 . . . . . . . . . 10 (𝑔 = 𝑛 → (𝑔𝑎) = (𝑛𝑎))
54cbvmptv 5261 . . . . . . . . 9 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)) = (𝑛 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑛𝑎))
63, 5fsetfocdm 8900 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵)
7 focdmex 7979 . . . . . . . 8 ({𝑓𝑓:𝐴𝐵} ∈ V → ((𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵𝐵 ∈ V))
86, 7syl5com 31 . . . . . . 7 ((𝐴𝑉𝑎𝐴) → ({𝑓𝑓:𝐴𝐵} ∈ V → 𝐵 ∈ V))
98nelcon3d 3048 . . . . . 6 ((𝐴𝑉𝑎𝐴) → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V))
109expcom 413 . . . . 5 (𝑎𝐴 → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
1110exlimiv 1928 . . . 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 1776  wcel 2106  {cab 2712  wne 2938  wnel 3044  Vcvv 3478  c0 4339  cmpt 5231  wf 6559  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  fsetcdmex  8902  fsetexb  8903
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