Theorem ac6 3576

Description: Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a large set B, where φ depends on x (the natural number) and y (to specify a member of B). A stronger version of this theorem, ac6s 3577, allows B to be a proper class.

Hypotheses

Ref	Expression
ac6.1	⊢ A ∈ V
ac6.2	⊢ B ∈ V
ac6.3	⊢ (y = (f ‘x) → (φ ↔ ψ))

Assertion

Ref	Expression
ac6	⊢ (∀x ∈ A ∃y ∈ B φ → ∃f(f:A–→B ∧ ∀x ∈ A ψ))

Distinct variable group(s):   x,f,y,A   B,f,x,y   φ,f   ψ,y

Proof of Theorem ac6

Step	Hyp	Ref	Expression
1		ac6.1	. . 3 ⊢ A ∈ V
2		ac6.2	. . 3 ⊢ B ∈ V
3		cleqid 1102	. . 3 ⊢ {y ∈ B∣φ} = {y ∈ B∣φ}
4		cleq1 1107	. . . . 5 ⊢ (w = z → (w = {y ∈ B∣φ} ↔ z = {y ∈ B∣φ}))
5	4	anbi2d 468	. . . 4 ⊢ (w = z → ((x ∈ A ∧ w = {y ∈ B∣φ}) ↔ (x ∈ A ∧ z = {y ∈ B∣φ})))
6	5	cbvopab2v 2109	. . 3 ⊢ {⟨x, w⟩∣(x ∈ A ∧ w = {y ∈ B∣φ})} = {⟨x, z⟩∣(x ∈ A ∧ z = {y ∈ B∣φ})}
7	1, 2, 3, 6	ac6lem 3575	. 2 ⊢ (∀x ∈ A ∃y ∈ B φ → ∃f(f:A–→B ∧ ∀x ∈ A (f ‘x) ∈ {y ∈ B∣φ}))
8		ac6.3	. . . . . . 7 ⊢ (y = (f ‘x) → (φ ↔ ψ))
9	8	elrab 1422	. . . . . 6 ⊢ ((f ‘x) ∈ {y ∈ B∣φ} ↔ ((f ‘x) ∈ B ∧ ψ))
10	9	pm3.27bd 263	. . . . 5 ⊢ ((f ‘x) ∈ {y ∈ B∣φ} → ψ)
11	10	r19.20si 1254	. . . 4 ⊢ (∀x ∈ A (f ‘x) ∈ {y ∈ B∣φ} → ∀x ∈ A ψ)
12	11	anim2i 270	. . 3 ⊢ ((f:A–→B ∧ ∀x ∈ A (f ‘x) ∈ {y ∈ B∣φ}) → (f:A–→B ∧ ∀x ∈ A ψ))
13	12	19.22i 723	. 2 ⊢ (∃f(f:A–→B ∧ ∀x ∈ A (f ‘x) ∈ {y ∈ B∣φ}) → ∃f(f:A–→B ∧ ∀x ∈ A ψ))
14	7, 13	syl 12	1 ⊢ (∀x ∈ A ∃y ∈ B φ → ∃f(f:A–→B ∧ ∀x ∈ A ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348  {copab 2055  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  ac6s 3577  projlem17 5209  osumlem5 5534

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-f 2434  df-fv 2438

metamath.org