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Theorem fsetprcnex 8803
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 8796 for 𝐴 ∉ V, fset0 8795 for 𝐴 = ∅, and fsetex 8797 for 𝐵 ∈ V, see also fsetexb 8805. (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 4307 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
2 feq1 6650 . . . . . . . . . 10 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
32cbvabv 2806 . . . . . . . . 9 {𝑓𝑓:𝐴𝐵} = {𝑚𝑚:𝐴𝐵}
4 fveq1 6842 . . . . . . . . . 10 (𝑔 = 𝑛 → (𝑔𝑎) = (𝑛𝑎))
54cbvmptv 5219 . . . . . . . . 9 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)) = (𝑛 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑛𝑎))
63, 5fsetfocdm 8802 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵)
7 focdmex 7889 . . . . . . . 8 ({𝑓𝑓:𝐴𝐵} ∈ V → ((𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↦ (𝑔𝑎)):{𝑓𝑓:𝐴𝐵}–onto𝐵𝐵 ∈ V))
86, 7syl5com 31 . . . . . . 7 ((𝐴𝑉𝑎𝐴) → ({𝑓𝑓:𝐴𝐵} ∈ V → 𝐵 ∈ V))
98nelcon3d 3058 . . . . . 6 ((𝐴𝑉𝑎𝐴) → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V))
109expcom 415 . . . . 5 (𝑎𝐴 → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
1110exlimiv 1934 . . . 4 (∃𝑎 𝑎𝐴 → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
121, 11sylbi 216 . . 3 (𝐴 ≠ ∅ → (𝐴𝑉 → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V)))
1312impcom 409 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓𝑓:𝐴𝐵} ∉ V))
1413imp 408 1 (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  {cab 2710  wne 2940  wnel 3046  Vcvv 3444  c0 4283  cmpt 5189  wf 6493  ontowfo 6495  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  fsetcdmex  8804  fsetexb  8805
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