Theorem fsetprcnexALT 44556

Description: First version of proof for fsetprcnex 8650, which was much more complicated. (Contributed by AV, 14-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fsetprcnexALT	⊢ (((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 fsetprcnexALT

Dummy variables 𝑎 𝑏 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abanssl 4235	. 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
2		n0 4280	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
3		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑦 ∈ V)
5		fsetsnprcnex 44549	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∉ V)
6	4, 5	sylan 580	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∉ V)
7		df-nel 3050	. . . . . . . . . 10 ⊢ ({𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∉ V ↔ ¬ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∈ V)
8	6, 7	sylib 217	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∉ V) → ¬ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∈ V)
9		eqid 2738	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
10		eqid 2738	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} = {𝑓 ∣ 𝑓:{𝑦}⟶𝐵}
11		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))) = (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦)))
12	9, 10, 11	cfsetsnfsetf1o 44555	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))):{𝑓 ∣ 𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)})
13	12	ancoms 459	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))):{𝑓 ∣ 𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)})
14	13	adantr 481	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∉ V) → (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))):{𝑓 ∣ 𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)})
15		f1ovv 7800	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))):{𝑓 ∣ 𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} → ({𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∈ V ↔ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V))
16	15	bicomd 222	. . . . . . . . . 10 ⊢ ((𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))):{𝑓 ∣ 𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} → ({𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V ↔ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∈ V))
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∉ V) → ({𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V ↔ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∈ V))
18	8, 17	mtbird 325	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∉ V) → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)
19	18	exp31 420	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)))
20	19	exlimiv 1933	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)))
21	2, 20	sylbi 216	. . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)))
22	21	impcom 408	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V))
23	22	imp 407	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)
24		df-nel 3050	. . 3 ⊢ ({𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∉ V ↔ ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)
25	23, 24	sylibr 233	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∉ V)
26		prcssprc 5249	. 2 ⊢ (({𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∧ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
27	1, 25, 26	sylancr 587	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943   ∉ wnel 3049  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  {csn 4561   ↦ cmpt 5157  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by: (None)