Theorem gblacfnacd 35075

Description: If 𝐹 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10191) holds. Note that 𝐹 must be a proper class by fndmexb 7946. 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 1929 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 6679	. . . 4 ⊢ (𝐹 Fn V → Fun 𝐹)
3		resfunexg 7252	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
4	3	elvd 3494	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝑥) ∈ V)
5	1, 2, 4	3syl 18	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑥) ∈ V)
6		ssv 4033	. . . . 5 ⊢ 𝑥 ⊆ V
7		fnssres 6703	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝑥 ⊆ V) → (𝐹 ↾ 𝑥) Fn 𝑥)
8	1, 6, 7	sylancl 585	. . . 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 6939	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝐹 ↾ 𝑥)‘𝑧) = (𝐹‘𝑧))
12	11	eleq1d 2829	. . . . . . 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 3151	. . . 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 6670	. . . 4 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑓 Fn 𝑥 ↔ (𝐹 ↾ 𝑥) Fn 𝑥))
18		fveq1 6919	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑓‘𝑧) = ((𝐹 ↾ 𝑥)‘𝑧))
19	18	eleq1d 2829	. . . . . 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 3184	. . . 4 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐹 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
22	17, 21	anbi12d 631	. . 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 3611	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
24	23	alrimiv 1926	1 ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   ↾ cres 5702  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by: (None)