Theorem ac5 3573

Description: An Axiom of Choice equivalent. Axiom AC of [BellMachover] p. 488.

Hypothesis

Ref	Expression
ac5.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,f,A

Proof of Theorem ac5

Step	Hyp	Ref	Expression
1		ac5.1	. 2 ⊢ A ∈ V
2		fneq2 2719	. . . 4 ⊢ (y = A → (f Fn y ↔ f Fn A))
3		raleq 1324	. . . 4 ⊢ (y = A → (∀x ∈ y (¬ x = ∅ → (f ‘x) ∈ x) ↔ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x)))
4	2, 3	anbi12d 476	. . 3 ⊢ (y = A → ((f Fn y ∧ ∀x ∈ y (¬ x = ∅ → (f ‘x) ∈ x)) ↔ (f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))))
5	4	biexdv 936	. 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 3556	. . . . 5 ⊢ (∀y∃f(f ⊆ y ∧ f Fn dom y) ↔ ∀y∃f∀x ∈ y (¬ x = ∅ → (f ‘x) ∈ x))
7		ac4 3571	. . . . 5 ⊢ ∃f∀x ∈ y (¬ x = ∅ → (f ‘x) ∈ x)
8	6, 7	mpgbir 686	. . . 4 ⊢ ∀y∃f(f ⊆ y ∧ f Fn dom y)
9		aceq4 3557	. . . 4 ⊢ (∀y∃f(f ⊆ y ∧ f Fn dom y) ↔ ∀y∃f(f Fn y ∧ ∀x ∈ y (¬ x = ∅ → (f ‘x) ∈ x)))
10	8, 9	mpbi 164	. . 3 ⊢ ∀y∃f(f Fn y ∧ ∀x ∈ y (¬ x = ∅ → (f ‘x) ∈ x))
11	10	a4i 680	. 2 ⊢ ∃f(f Fn y ∧ ∀x ∈ y (¬ x = ∅ → (f ‘x) ∈ x))
12	1, 5, 11	vtocl 1378	1 ⊢ ∃f(f Fn A ∧ ∀x ∈ A (¬ x = ∅ → (f ‘x) ∈ x))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  dom cdm 2410   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  ac5b 3574

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-fr 2169  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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