Theorem aceq4 3557

Description: Equivalence of two versions of the Axiom of Choice. The left-hand side is similar to the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC.

Assertion

Ref	Expression
aceq4	⊢ (∀x∃f(f ⊆ x ∧ f Fn dom x) ↔ ∀x∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))

Distinct variable group(s):   x,f,z

Proof of Theorem aceq4

Step	Hyp	Ref	Expression
1		aceq3 3556	. 2 ⊢ (∀x∃f(f ⊆ x ∧ f Fn dom x) ↔ ∀x∃f∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z))
2		fveq1 2831	. . . . . . . . 9 ⊢ (f = y → (f ‘z) = (y ‘z))
3	2	eleq1d 1155	. . . . . . . 8 ⊢ (f = y → ((f ‘z) ∈ z ↔ (y ‘z) ∈ z))
4	3	imbi2d 464	. . . . . . 7 ⊢ (f = y → ((¬ z = ∅ → (f ‘z) ∈ z) ↔ (¬ z = ∅ → (y ‘z) ∈ z)))
5	4	biraldv 1219	. . . . . 6 ⊢ (f = y → (∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z) ↔ ∀z ∈ x (¬ z = ∅ → (y ‘z) ∈ z)))
6	5	cbvexv 973	. . . . 5 ⊢ (∃f∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z) ↔ ∃y∀z ∈ x (¬ z = ∅ → (y ‘z) ∈ z))
7		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (w = z → (y ‘w) = (y ‘z))
8		cleqid 1102	. . . . . . . . . . . . 13 ⊢ {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} = {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))}
9		fvex 2838	. . . . . . . . . . . . 13 ⊢ (y ‘z) ∈ V
10	7, 8, 9	fvopab4 2871	. . . . . . . . . . . 12 ⊢ (z ∈ x → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) = (y ‘z))
11	10	eleq1d 1155	. . . . . . . . . . 11 ⊢ (z ∈ x → (({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z ↔ (y ‘z) ∈ z))
12	11	imbi2d 464	. . . . . . . . . 10 ⊢ (z ∈ x → ((¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z) ↔ (¬ z = ∅ → (y ‘z) ∈ z)))
13	12	birala 1228	. . . . . . . . 9 ⊢ (∀z ∈ x (¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z) ↔ ∀z ∈ x (¬ z = ∅ → (y ‘z) ∈ z))
14	13	anbi2i 367	. . . . . . . 8 ⊢ (({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} Fn x ∧ ∀z ∈ x (¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z)) ↔ ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} Fn x ∧ ∀z ∈ x (¬ z = ∅ → (y ‘z) ∈ z)))
15		fvex 2838	. . . . . . . . 9 ⊢ (y ‘w) ∈ V
16	15, 8	fnopab2 2747	. . . . . . . 8 ⊢ {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} Fn x
17	14, 16	mpbiran 547	. . . . . . 7 ⊢ (({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} Fn x ∧ ∀z ∈ x (¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z)) ↔ ∀z ∈ x (¬ z = ∅ → (y ‘z) ∈ z))
18		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
19		moeq 1431	. . . . . . . . . 10 ⊢ ∃*v v = (y ‘w)
20	19	a1i 7	. . . . . . . . 9 ⊢ (w ∈ x → ∃*v v = (y ‘w))
21	18, 20	funopabex 2742	. . . . . . . 8 ⊢ {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ∈ V
22		fneq1 2718	. . . . . . . . 9 ⊢ (f = {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} → (f Fn x ↔ {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} Fn x))
23		fveq1 2831	. . . . . . . . . . . 12 ⊢ (f = {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} → (f ‘z) = ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z))
24	23	eleq1d 1155	. . . . . . . . . . 11 ⊢ (f = {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} → ((f ‘z) ∈ z ↔ ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z))
25	24	imbi2d 464	. . . . . . . . . 10 ⊢ (f = {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} → ((¬ z = ∅ → (f ‘z) ∈ z) ↔ (¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z)))
26	25	biraldv 1219	. . . . . . . . 9 ⊢ (f = {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} → (∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z) ↔ ∀z ∈ x (¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z)))
27	22, 26	anbi12d 476	. . . . . . . 8 ⊢ (f = {⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} → ((f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)) ↔ ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} Fn x ∧ ∀z ∈ x (¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z))))
28	21, 27	cla4ev 1401	. . . . . . 7 ⊢ (({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} Fn x ∧ ∀z ∈ x (¬ z = ∅ → ({⟨w, v⟩∣(w ∈ x ∧ v = (y ‘w))} ‘z) ∈ z)) → ∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))
29	17, 28	sylbir 176	. . . . . 6 ⊢ (∀z ∈ x (¬ z = ∅ → (y ‘z) ∈ z) → ∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))
30	29	19.23aiv 952	. . . . 5 ⊢ (∃y∀z ∈ x (¬ z = ∅ → (y ‘z) ∈ z) → ∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))
31	6, 30	sylbi 174	. . . 4 ⊢ (∃f∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z) → ∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))
32		pm3.27 260	. . . . 5 ⊢ ((f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)) → ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z))
33	32	19.22i 723	. . . 4 ⊢ (∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)) → ∃f∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z))
34	31, 33	impbi 139	. . 3 ⊢ (∃f∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z) ↔ ∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))
35	34	bial 695	. 2 ⊢ (∀x∃f∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z) ↔ ∀x∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))
36	1, 35	bitr 151	1 ⊢ (∀x∃f(f ⊆ x ∧ f Fn dom x) ↔ ∀x∃f(f Fn x ∧ ∀z ∈ x (¬ z = ∅ → (f ‘z) ∈ z)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ∅c0 1707  {copab 2055  dom cdm 2410   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  aceq5 3563  ac5 3573

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

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-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-id 2125  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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