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Theorem fsetprcnex 8879
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 8872 for 𝐴 ∉ V, fset0 8871 for 𝐴 = ∅, and fsetex 8873 for 𝐵 ∈ V, see also fsetexb 8881. (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 4342 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
2 feq1 6698 . . . . . . . . . 10 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
32cbvabv 2798 . . . . . . . . 9 {𝑓𝑓:𝐴𝐵} = {𝑚𝑚:𝐴𝐵}
4 fveq1 6891 . . . . . . . . . 10 (𝑔 = 𝑛 → (𝑔𝑎) = (𝑛𝑎))
54cbvmptv 5256 . . . . . . . . 9 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)) = (𝑛 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑛𝑎))
63, 5fsetfocdm 8878 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵)
7 focdmex 7957 . . . . . . . 8 ({𝑓𝑓:𝐴𝐵} ∈ V → ((𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵𝐵 ∈ V))
86, 7syl5com 31 . . . . . . 7 ((𝐴𝑉𝑎𝐴) → ({𝑓𝑓:𝐴𝐵} ∈ V → 𝐵 ∈ V))
98nelcon3d 3040 . . . . . 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 2702  wne 2930  wnel 3036  Vcvv 3463  c0 4318  cmpt 5226  wf 6539  ontowfo 6541  cfv 6543
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  fsetcdmex  8880  fsetexb  8881
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