Theorem gblacfnac2d 35949

Description: If 𝐺 is a global choice function, then the Axiom of Choice holds. This is gblacfnacd 35948 expressed using CHOICE. (Contributed by BTernaryTau, 25-Sep-2026.)

Hypotheses

Ref	Expression
gblacfnac2d.1	⊢ (𝜑 → 𝐺 Fn V)
gblacfnac2d.2	⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))

Assertion

Ref	Expression
gblacfnac2d	⊢ (𝜑 → CHOICE)

Distinct variable groups:   𝜑,𝑧   𝑧,𝐺

Proof of Theorem gblacfnac2d

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gblacfnac2d.1	. . 3 ⊢ (𝜑 → 𝐺 Fn V)
2		gblacfnac2d.2	. . 3 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
3	1, 2	gblacfnacd 35948	. 2 ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
4		dfac4 10204	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
5	3, 4	sylibr 237	1 ⊢ (𝜑 → CHOICE)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4278   Fn wfn 6533  ‘cfv 6538  CHOICEwac 10197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ac 10198

This theorem is used by: (None)