Theorem gblacfnacd 35135

Description: If 𝐹 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10141) holds. Note that 𝐹 must be a proper class by fndmexb 7907. This means we cannot show that the existence of a class that behaves as a global choice function is sufficient because we only have existential quantifiers for sets, not (proper) classes. However, if a class variant of exlimiv 1930 were available, then it could be used alongside the closed form of this theorem to prove that result. (Contributed by BTernaryTau, 12-Dec-2024.)

Hypotheses

Ref	Expression
gblacfnacd.1	⊢ (𝜑 → 𝐹 Fn V)
gblacfnacd.2	⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))

Assertion

Ref	Expression
gblacfnacd	⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))

Distinct variable groups:   𝑥,𝑓   𝜑,𝑥,𝑧   𝑓,𝐹,𝑧

Allowed substitution hints:   𝜑(𝑓)   𝐹(𝑥)

Proof of Theorem gblacfnacd

Step	Hyp	Ref	Expression
1		gblacfnacd.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn V)
2		fnfun 6643	. . . 4 ⊢ (𝐹 Fn V → Fun 𝐹)
3		resfunexg 7212	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
4	3	elvd 3470	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝑥) ∈ V)
5	1, 2, 4	3syl 18	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑥) ∈ V)
6		ssv 3988	. . . . 5 ⊢ 𝑥 ⊆ V
7		fnssres 6666	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝑥 ⊆ V) → (𝐹 ↾ 𝑥) Fn 𝑥)
8	1, 6, 7	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑥) Fn 𝑥)
9		gblacfnacd.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
10	9	19.21bi 2190	. . . . . 6 ⊢ (𝜑 → (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
11		fvres 6900	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝐹 ↾ 𝑥)‘𝑧) = (𝐹‘𝑧))
12	11	eleq1d 2820	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → (((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧 ↔ (𝐹‘𝑧) ∈ 𝑧))
13	12	imbi2d 340	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)))
14	10, 13	syl5ibrcom 247	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
15	14	ralrimiv 3132	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧))
16	8, 15	jca 511	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
17		fneq1 6634	. . . 4 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑓 Fn 𝑥 ↔ (𝐹 ↾ 𝑥) Fn 𝑥))
18		fveq1 6880	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑓‘𝑧) = ((𝐹 ↾ 𝑥)‘𝑧))
19	18	eleq1d 2820	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧))
20	19	imbi2d 340	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
21	20	ralbidv 3164	. . . 4 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
22	17, 21	anbi12d 632	. . 3 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ((𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ↔ ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧))))
23	5, 16, 22	spcedv 3582	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
24	23	alrimiv 1927	1 ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  ∅c0 4313   ↾ cres 5661  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by: (None)