Theorem fsetprcnexALT 47012

Description: First version of proof for fsetprcnex 8901, 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 4317	. 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
2		n0 4359	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
3		vex 3482	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑦 ∈ V)
5		fsetsnprcnex 47005	. . . . . . . . . . 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 3045	. . . . . . . . . 10 ⊢ ({𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∉ V ↔ ¬ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∈ V)
8	6, 7	sylib 218	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∉ V) → ¬ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∈ V)
9		eqid 2735	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} = {𝑓 ∣ 𝑓:{𝑦}⟶𝐵}
11		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))) = (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦)))
12	9, 10, 11	cfsetsnfsetf1o 47011	. . . . . . . . . . . 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 458	. . . . . . . . . . 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 480	. . . . . . . . . 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 7981	. . . . . . . . . . 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 223	. . . . . . . . . 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 419	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)))
20	19	exlimiv 1928	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)))
21	2, 20	sylbi 217	. . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)))
22	21	impcom 407	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V))
23	22	imp 406	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)
24		df-nel 3045	. . 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 234	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∉ V)
26		prcssprc 5333	. 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 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938   ∉ wnel 3044  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ⊆ wss 3963  ∅c0 4339  {csn 4631   ↦ cmpt 5231  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by: (None)