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Theorem fsetprcnex 8887
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 8880 for 𝐴 ∉ V, fset0 8879 for 𝐴 = ∅, and fsetex 8881 for 𝐵 ∈ V, see also fsetexb 8889. (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 4350 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
2 feq1 6708 . . . . . . . . . 10 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
32cbvabv 2801 . . . . . . . . 9 {𝑓𝑓:𝐴𝐵} = {𝑚𝑚:𝐴𝐵}
4 fveq1 6901 . . . . . . . . . 10 (𝑔 = 𝑛 → (𝑔𝑎) = (𝑛𝑎))
54cbvmptv 5265 . . . . . . . . 9 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)) = (𝑛 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑛𝑎))
63, 5fsetfocdm 8886 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵)
7 focdmex 7965 . . . . . . . 8 ({𝑓𝑓:𝐴𝐵} ∈ V → ((𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵𝐵 ∈ V))
86, 7syl5com 31 . . . . . . 7 ((𝐴𝑉𝑎𝐴) → ({𝑓𝑓:𝐴𝐵} ∈ V → 𝐵 ∈ V))
98nelcon3d 3047 . . . . . 6 ((𝐴𝑉𝑎𝐴) → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V))
109expcom 412 . . . . 5 (𝑎𝐴 → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
1110exlimiv 1925 . . . 4 (∃𝑎 𝑎𝐴 → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
121, 11sylbi 216 . . 3 (𝐴 ≠ ∅ → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
1312impcom 406 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V))
1413imp 405 1 (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wex 1773  wcel 2098  {cab 2705  wne 2937  wnel 3043  Vcvv 3473  c0 4326  cmpt 5235  wf 6549  ontowfo 6551  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561
This theorem is referenced by:  fsetcdmex  8888  fsetexb  8889
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