Theorem ac5 4744

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 4742.

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

Step	Hyp	Ref	Expression
1		ac5.1	. 2 ⊢ A ∈ V
2		fneq2 3586	. . . 4 ⊢ (y = A → (f Fn y ↔ f Fn A))
3		raleq1 1783	. . . 4 ⊢ (y = A → (∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x) ↔ ∀x ∈ A (x ≠ ∅ → (f ‘x) ∈ x)))
4	2, 3	anbi12d 627	. . 3 ⊢ (y = A → ((f Fn y ⋀ ∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x)) ↔ (f Fn A ⋀ ∀x ∈ A (x ≠ ∅ → (f ‘x) ∈ x))))
5	4	exbidv 1277	. 2 ⊢ (y = A → (∃f(f Fn y ⋀ ∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x)) ↔ ∃f(f Fn A ⋀ ∀x ∈ A (x ≠ ∅ → (f ‘x) ∈ x))))
6		aceq3 4725	. . . . 5 ⊢ (∀y∃f(f ⊆ y ⋀ f Fn dom y) ↔ ∀y∃f∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x))
7		ac4 4742	. . . . 5 ⊢ ∃f∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x)
8	6, 7	mpgbir 986	. . . 4 ⊢ ∀y∃f(f ⊆ y ⋀ f Fn dom y)
9		aceq4 4726	. . . 4 ⊢ (∀y∃f(f ⊆ y ⋀ f Fn dom y) ↔ ∀y∃f(f Fn y ⋀ ∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x)))
10	8, 9	mpbi 189	. . 3 ⊢ ∀y∃f(f Fn y ⋀ ∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x))
11	10	a4i 980	. 2 ⊢ ∃f(f Fn y ⋀ ∀x ∈ y (x ≠ ∅ → (f ‘x) ∈ x))
12	1, 5, 11	vtocl 1838	1 ⊢ ∃f(f Fn A ⋀ ∀x ∈ A (x ≠ ∅ → (f ‘x) ∈ x))

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978   ≠ wne 1582  ∀wral 1642  Vcvv 1807   ⊆ wss 2043  ∅c0 2276  dom cdm 3170   Fn wfn 3177   ‘cfv 3182

This theorem is referenced by:  ac5b 4745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2865  ax-ac 4736

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2500  df-br 2616  df-opab 2663  df-id 2832  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

Copyright terms: Public domain