Theorem gblacfnacd 35218

Description: If 𝐺 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10024) holds. Note that 𝐺 must be a proper class by fndmexb 7845. 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 1931 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 → 𝐺 Fn V)
gblacfnacd.2	⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘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,𝐺,𝑧

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

Proof of Theorem gblacfnacd

Step	Hyp	Ref	Expression
1		gblacfnacd.1	. . . 4 ⊢ (𝜑 → 𝐺 Fn V)
2		fnfun 6589	. . . 4 ⊢ (𝐺 Fn V → Fun 𝐺)
3		resfunexg 7158	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ V) → (𝐺 ↾ 𝑥) ∈ V)
4	3	elvd 3443	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ↾ 𝑥) ∈ V)
5	1, 2, 4	3syl 18	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝑥) ∈ V)
6		ssv 3955	. . . . 5 ⊢ 𝑥 ⊆ V
7		fnssres 6612	. . . . 5 ⊢ ((𝐺 Fn V ∧ 𝑥 ⊆ V) → (𝐺 ↾ 𝑥) Fn 𝑥)
8	1, 6, 7	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝑥) Fn 𝑥)
9		gblacfnacd.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
10	9	19.21bi 2194	. . . . . 6 ⊢ (𝜑 → (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
11		fvres 6850	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝐺 ↾ 𝑥)‘𝑧) = (𝐺‘𝑧))
12	11	eleq1d 2818	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → (((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧 ↔ (𝐺‘𝑧) ∈ 𝑧))
13	12	imbi2d 340	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧)))
14	10, 13	syl5ibrcom 247	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
15	14	ralrimiv 3124	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧))
16	8, 15	jca 511	. . 3 ⊢ (𝜑 → ((𝐺 ↾ 𝑥) Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
17		fneq1 6580	. . . 4 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓 Fn 𝑥 ↔ (𝐺 ↾ 𝑥) Fn 𝑥))
18		fveq1 6830	. . . . . . 7 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓‘𝑧) = ((𝐺 ↾ 𝑥)‘𝑧))
19	18	eleq1d 2818	. . . . . 6 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧))
20	19	imbi2d 340	. . . . 5 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
21	20	ralbidv 3156	. . . 4 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
22	17, 21	anbi12d 632	. . 3 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → ((𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ↔ ((𝐺 ↾ 𝑥) Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧))))
23	5, 16, 22	spcedv 3549	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
24	23	alrimiv 1928	1 ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  Vcvv 3437   ⊆ wss 3898  ∅c0 4282   ↾ cres 5623  Fun wfun 6483   Fn wfn 6484  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497

This theorem is referenced by: (None)