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Theorem ac5 10517
Description: An Axiom of Choice equivalent: there exists a function 𝑓 (called a choice function) with domain 𝐴 that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that 𝑓 be a function is not necessary; see ac4 10515. (Contributed by NM, 29-Aug-1999.)
Hypothesis
Ref Expression
ac5.1 𝐴 ∈ V
Assertion
Ref Expression
ac5 𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))
Distinct variable group:   𝑥,𝑓,𝐴

Proof of Theorem ac5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ac5.1 . 2 𝐴 ∈ V
2 fneq2 6660 . . . 4 (𝑦 = 𝐴 → (𝑓 Fn 𝑦𝑓 Fn 𝐴))
3 raleq 3323 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥) ↔ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
42, 3anbi12d 632 . . 3 (𝑦 = 𝐴 → ((𝑓 Fn 𝑦 ∧ ∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))))
54exbidv 1921 . 2 (𝑦 = 𝐴 → (∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)) ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))))
6 dfac4 10162 . . 3 (CHOICE ↔ ∀𝑦𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
76axaci 10508 . 2 𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))
81, 5, 7vtocl 3558 1 𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  c0 4333   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-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503
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-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-pw 4602  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-fv 6569  df-ac 10156
This theorem is referenced by: (None)
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