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Theorem gblacfnacd 35113
Description: If 𝐹 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10162) holds. Note that 𝐹 must be a proper class by fndmexb 7928. 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 (𝜑𝐹 Fn V)
gblacfnacd.2 (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
Assertion
Ref Expression
gblacfnacd (𝜑 → ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
Distinct variable groups:   𝑥,𝑓   𝜑,𝑥,𝑧   𝑓,𝐹,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝐹(𝑥)

Proof of Theorem gblacfnacd
StepHypRef Expression
1 gblacfnacd.1 . . . 4 (𝜑𝐹 Fn V)
2 fnfun 6668 . . . 4 (𝐹 Fn V → Fun 𝐹)
3 resfunexg 7235 . . . . 5 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
43elvd 3486 . . . 4 (Fun 𝐹 → (𝐹𝑥) ∈ V)
51, 2, 43syl 18 . . 3 (𝜑 → (𝐹𝑥) ∈ V)
6 ssv 4008 . . . . 5 𝑥 ⊆ V
7 fnssres 6691 . . . . 5 ((𝐹 Fn V ∧ 𝑥 ⊆ V) → (𝐹𝑥) Fn 𝑥)
81, 6, 7sylancl 586 . . . 4 (𝜑 → (𝐹𝑥) Fn 𝑥)
9 gblacfnacd.2 . . . . . . 7 (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
10919.21bi 2189 . . . . . 6 (𝜑 → (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
11 fvres 6925 . . . . . . . 8 (𝑧𝑥 → ((𝐹𝑥)‘𝑧) = (𝐹𝑧))
1211eleq1d 2826 . . . . . . 7 (𝑧𝑥 → (((𝐹𝑥)‘𝑧) ∈ 𝑧 ↔ (𝐹𝑧) ∈ 𝑧))
1312imbi2d 340 . . . . . 6 (𝑧𝑥 → ((𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧)))
1410, 13syl5ibrcom 247 . . . . 5 (𝜑 → (𝑧𝑥 → (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧)))
1514ralrimiv 3145 . . . 4 (𝜑 → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧))
168, 15jca 511 . . 3 (𝜑 → ((𝐹𝑥) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧)))
17 fneq1 6659 . . . 4 (𝑓 = (𝐹𝑥) → (𝑓 Fn 𝑥 ↔ (𝐹𝑥) Fn 𝑥))
18 fveq1 6905 . . . . . . 7 (𝑓 = (𝐹𝑥) → (𝑓𝑧) = ((𝐹𝑥)‘𝑧))
1918eleq1d 2826 . . . . . 6 (𝑓 = (𝐹𝑥) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝐹𝑥)‘𝑧) ∈ 𝑧))
2019imbi2d 340 . . . . 5 (𝑓 = (𝐹𝑥) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧)))
2120ralbidv 3178 . . . 4 (𝑓 = (𝐹𝑥) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧)))
2217, 21anbi12d 632 . . 3 (𝑓 = (𝐹𝑥) → ((𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ ((𝐹𝑥) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧))))
235, 16, 22spcedv 3598 . 2 (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
2423alrimiv 1927 1 (𝜑 → ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  wss 3951  c0 4333  cres 5687  Fun wfun 6555   Fn wfn 6556  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
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