Theorem fsetprcnex 8859

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 8852 for 𝐴 ∉ V, fset0 8851 for 𝐴 = ∅, and fsetex 8853 for 𝐵 ∈ V, see also fsetexb 8861. (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 4315	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐴)
2		feq1 6684	. . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
3	2	cbvabv 2839	. . . . . . . . 9 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} = {𝑚 ∣ 𝑚:𝐴⟶𝐵}
4		fveq1 6881	. . . . . . . . . 10 ⊢ (𝑔 = 𝑛 → (𝑔‘𝑎) = (𝑛‘𝑎))
5	4	cbvmptv 5219	. . . . . . . . 9 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)) = (𝑛 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑛‘𝑎))
6	3, 5	fsetfocdm 8858	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↦ (𝑔‘𝑎)):{𝑓 ∣ 𝑓:𝐴⟶𝐵}–onto→𝐵)
7		focdmex 7953	. . . . . . . 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 32	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V → 𝐵 ∈ V))
9	8	nelcon3d 3074	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))
10	9	expcom 418	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)))
11	10	exlimiv 1957	. . . 4 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)))
12	1, 11	sylbi 220	. . 3 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)))
13	12	impcom 412	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∉ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V))
14	13	imp 411	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  {cab 2747   ≠ wne 2964   ∉ wnel 3070  Vcvv 3463  ∅c0 4294   ↦ cmpt 5196  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  fsetcdmex  8860  fsetexb  8861