Theorem gblacfnacd 35586

Description: If 𝐺 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10102) holds. Note that 𝐺 must be a proper class by fndmexb 7899. 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 1960 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 6635	. . . 4 ⊢ (𝐺 Fn V → Fun 𝐺)
3		resfunexg 7213	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ V) → (𝐺 ↾ 𝑥) ∈ V)
4	3	elvd 3461	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ↾ 𝑥) ∈ V)
5	1, 2, 4	3syl 19	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝑥) ∈ V)
6		ssv 3961	. . . . 5 ⊢ 𝑥 ⊆ V
7		fnssres 6658	. . . . 5 ⊢ ((𝐺 Fn V ∧ 𝑥 ⊆ V) → (𝐺 ↾ 𝑥) Fn 𝑥)
8	1, 6, 7	sylancl 597	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝑥) Fn 𝑥)
9		gblacfnacd.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
10	9	19.21bi 2225	. . . . . 6 ⊢ (𝜑 → (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
11		fvres 6900	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝐺 ↾ 𝑥)‘𝑧) = (𝐺‘𝑧))
12	11	eleq1d 2848	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → (((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧 ↔ (𝐺‘𝑧) ∈ 𝑧))
13	12	imbi2d 343	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧)))
14	10, 13	syl5ibrcom 250	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
15	14	ralrimiv 3156	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧))
16	8, 15	jca 520	. . 3 ⊢ (𝜑 → ((𝐺 ↾ 𝑥) Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
17		fneq1 6626	. . . 4 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓 Fn 𝑥 ↔ (𝐺 ↾ 𝑥) Fn 𝑥))
18		fveq1 6880	. . . . . . 7 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓‘𝑧) = ((𝐺 ↾ 𝑥)‘𝑧))
19	18	eleq1d 2848	. . . . . 6 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧))
20	19	imbi2d 343	. . . . 5 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
21	20	ralbidv 3188	. . . 4 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
22	17, 21	anbi12d 643	. . 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 3557	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
24	23	alrimiv 1957	1 ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ⊆ wss 3905  ∅c0 4286   ↾ cres 5663  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 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  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)