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Theorem gblacfnacd 35092
Description: If 𝐹 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10160) holds. Note that 𝐹 must be a proper class by fndmexb 7929. 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 1928 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 6669 . . . 4 (𝐹 Fn V → Fun 𝐹)
3 resfunexg 7235 . . . . 5 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
43elvd 3484 . . . 4 (Fun 𝐹 → (𝐹𝑥) ∈ V)
51, 2, 43syl 18 . . 3 (𝜑 → (𝐹𝑥) ∈ V)
6 ssv 4020 . . . . 5 𝑥 ⊆ V
7 fnssres 6692 . . . . 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 2187 . . . . . 6 (𝜑 → (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
11 fvres 6926 . . . . . . . 8 (𝑧𝑥 → ((𝐹𝑥)‘𝑧) = (𝐹𝑧))
1211eleq1d 2824 . . . . . . 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 3143 . . . 4 (𝜑 → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧))
168, 15jca 511 . . 3 (𝜑 → ((𝐹𝑥) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧)))
17 fneq1 6660 . . . 4 (𝑓 = (𝐹𝑥) → (𝑓 Fn 𝑥 ↔ (𝐹𝑥) Fn 𝑥))
18 fveq1 6906 . . . . . . 7 (𝑓 = (𝐹𝑥) → (𝑓𝑧) = ((𝐹𝑥)‘𝑧))
1918eleq1d 2824 . . . . . 6 (𝑓 = (𝐹𝑥) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝐹𝑥)‘𝑧) ∈ 𝑧))
2019imbi2d 340 . . . . 5 (𝑓 = (𝐹𝑥) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝐹𝑥)‘𝑧) ∈ 𝑧)))
2120ralbidv 3176 . . . 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 1925 1 (𝜑 → ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  Vcvv 3478  wss 3963  c0 4339  cres 5691  Fun wfun 6557   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by: (None)
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