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Theorem ac5 10514
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 10512. (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 3320 . . . 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 1918 . 2 (𝑦 = 𝐴 → (∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)) ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))))
6 dfac4 10159 . . 3 (CHOICE ↔ ∀𝑦𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
76axaci 10505 . 2 𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))
81, 5, 7vtocl 3557 1 𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  Vcvv 3477  c0 4338   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-ac2 10500
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ac 10153
This theorem is referenced by: (None)
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