Theorem gblacfnacd 35096

Description: If 𝐺 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10082) holds. Note that 𝐺 must be a proper class by fndmexb 7885. 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 → 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 6621	. . . 4 ⊢ (𝐺 Fn V → Fun 𝐺)
3		resfunexg 7192	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ V) → (𝐺 ↾ 𝑥) ∈ V)
4	3	elvd 3456	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ↾ 𝑥) ∈ V)
5	1, 2, 4	3syl 18	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝑥) ∈ V)
6		ssv 3974	. . . . 5 ⊢ 𝑥 ⊆ V
7		fnssres 6644	. . . . 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 2190	. . . . . 6 ⊢ (𝜑 → (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
11		fvres 6880	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝐺 ↾ 𝑥)‘𝑧) = (𝐺‘𝑧))
12	11	eleq1d 2814	. . . . . . 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 3125	. . . 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 6612	. . . 4 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓 Fn 𝑥 ↔ (𝐺 ↾ 𝑥) Fn 𝑥))
18		fveq1 6860	. . . . . . 7 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓‘𝑧) = ((𝐺 ↾ 𝑥)‘𝑧))
19	18	eleq1d 2814	. . . . . 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 3157	. . . 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 3567	. 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 2926  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ∅c0 4299   ↾ cres 5643  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by: (None)