Theorem fsetprcnexALT 46257

Description: First version of proof for fsetprcnex 8852, 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 4293	. 2 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
2		n0 4338	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
3		vex 3470	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑦 ∈ V)
5		fsetsnprcnex 46250	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∉ V)
6	4, 5	sylan 579	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ∉ V)
7		df-nel 3039	. . . . . . . . . 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 2724	. . . . . . . . . . . . 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 2724	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} = {𝑓 ∣ 𝑓:{𝑦}⟶𝐵}
11		eqid 2724	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦))) = (𝑔 ∈ {𝑓 ∣ 𝑓:{𝑦}⟶𝐵} ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑦)))
12	9, 10, 11	cfsetsnfsetf1o 46256	. . . . . . . . . . . 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 7937	. . . . . . . . . . 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 419	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∈ 𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} ∈ V)))
20	19	exlimiv 1925	. . . . . 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 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 3039	. . 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 5315	. 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 586	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2701   ≠ wne 2932   ∉ wnel 3038  ∀wral 3053  ∃wrex 3062  Vcvv 3466   ⊆ wss 3940  ∅c0 4314  {csn 4620   ↦ cmpt 5221  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541

This theorem is referenced by: (None)