Theorem ac5 10430

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 10428. (Contributed by NM, 29-Aug-1999.)

Hypothesis

Ref	Expression
ac5.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ac5	⊢ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))

Distinct variable group:   𝑥,𝑓,𝐴

Proof of Theorem ac5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ac5.1	. 2 ⊢ 𝐴 ∈ V
2		fneq2 6610	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓 Fn 𝑦 ↔ 𝑓 Fn 𝐴))
3		raleq 3296	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)))
4	2, 3	anbi12d 632	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑓 Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))))
5	4	exbidv 1921	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)) ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))))
6		dfac4 10075	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)))
7	6	axaci 10421	. 2 ⊢ ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))
8	1, 5, 7	vtocl 3524	1 ⊢ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3447  ∅c0 4296   Fn wfn 6506  ‘cfv 6511

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-ac2 10416

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ac 10069

This theorem is referenced by: (None)