Theorem fsetprcnex 8809

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 {𝑓 ∣ 𝑓:𝐴⟶𝐵} is a set, see fsetdmprc0 8802 for 𝐴 ∉ V, fset0 8801 for 𝐴 = ∅, and fsetex 8803 for 𝐵 ∈ V, see also fsetexb 8811. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.)

Assertion

Ref	Expression
fsetprcnex	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ 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.

Step	Hyp	Ref	Expression
1		n0 4293	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐴)
2		feq1 6646	. . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
3	2	cbvabv 2806	. . . . . . . . 9 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} = {𝑚 ∣ 𝑚:𝐴⟶𝐵}
4		fveq1 6839	. . . . . . . . . 10 ⊢ (𝑔 = 𝑛 → (𝑔‘𝑎) = (𝑛‘𝑎))
5	4	cbvmptv 5189	. . . . . . . . 9 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)) = (𝑛 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑛‘𝑎))
6	3, 5	fsetfocdm 8808	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)):{𝑓 ∣ 𝑓:𝐴⟶𝐵}–onto→𝐵)
7		focdmex 7909	. . . . . . . 8 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V → ((𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)):{𝑓 ∣ 𝑓:𝐴⟶𝐵}–onto→𝐵 → 𝐵 ∈ V))
8	6, 7	syl5com 31	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V → 𝐵 ∈ V))
9	8	nelcon3d 3040	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))
10	9	expcom 413	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)))
11	10	exlimiv 1932	. . . 4 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)))
12	1, 11	sylbi 217	. . 3 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)))
13	12	impcom 407	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))
14	13	imp 406	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1781   ∈ wcel 2114  {cab 2714   ≠ wne 2932   ∉ wnel 3036  Vcvv 3429  ∅c0 4273   ↦ cmpt 5166  ⟶wf 6494  –onto→wfo 6496  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506

This theorem is referenced by:  fsetcdmex  8810  fsetexb  8811