Theorem gblacfnacd 35318

Description: If 𝐺 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10044) holds. Note that 𝐺 must be a proper class by fndmexb 7858. 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 1932 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 6600	. . . 4 ⊢ (𝐺 Fn V → Fun 𝐺)
3		resfunexg 7171	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ V) → (𝐺 ↾ 𝑥) ∈ V)
4	3	elvd 3448	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ↾ 𝑥) ∈ V)
5	1, 2, 4	3syl 18	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝑥) ∈ V)
6		ssv 3960	. . . . 5 ⊢ 𝑥 ⊆ V
7		fnssres 6623	. . . . 5 ⊢ ((𝐺 Fn V ∧ 𝑥 ⊆ V) → (𝐺 ↾ 𝑥) Fn 𝑥)
8	1, 6, 7	sylancl 587	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝑥) Fn 𝑥)
9		gblacfnacd.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
10	9	19.21bi 2197	. . . . . 6 ⊢ (𝜑 → (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
11		fvres 6861	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝐺 ↾ 𝑥)‘𝑧) = (𝐺‘𝑧))
12	11	eleq1d 2822	. . . . . . 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 3129	. . . 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 6591	. . . 4 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓 Fn 𝑥 ↔ (𝐺 ↾ 𝑥) Fn 𝑥))
18		fveq1 6841	. . . . . . 7 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (𝑓‘𝑧) = ((𝐺 ↾ 𝑥)‘𝑧))
19	18	eleq1d 2822	. . . . . 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 3161	. . . 4 ⊢ (𝑓 = (𝐺 ↾ 𝑥) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝐺 ↾ 𝑥)‘𝑧) ∈ 𝑧)))
22	17, 21	anbi12d 633	. . 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 3554	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
24	23	alrimiv 1929	1 ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  ∅c0 4287   ↾ cres 5634  Fun wfun 6494   Fn wfn 6495  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508

This theorem is referenced by: (None)