Theorem ac5 10560

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 10558. (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 6634	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓 Fn 𝑦 ↔ 𝑓 Fn 𝐴))
3		raleq 3316	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)))
4	2, 3	anbi12d 644	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑓 Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))))
5	4	exbidv 1953	. 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 10206	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)))
7	6	axaci 10551	. 2 ⊢ ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))
8	1, 5, 7	vtocl 3520	1 ⊢ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4277   Fn wfn 6536  ‘cfv 6541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-ac2 10546

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ac 10200

This theorem is used by: (None)