Theorem cc4f 7331

Description: Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)

Hypotheses

Ref	Expression
cc4f.cc	⊢ (𝜑 → CCHOICE)
cc4f.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
cc4f.a	⊢ Ⅎ𝑛𝐴
cc4f.2	⊢ (𝜑 → 𝑁 ≈ ω)
cc4f.3	⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))
cc4f.m	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
cc4f	⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))

Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝑁,𝑛   𝜒,𝑥   𝜑,𝑓,𝑛   𝜓,𝑓   𝑥,𝑛

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑛)   𝜒(𝑓,𝑛)   𝐴(𝑛)   𝑁(𝑥)   𝑉(𝑥,𝑓,𝑛)

Proof of Theorem cc4f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cc4f.cc	. . 3 ⊢ (𝜑 → CCHOICE)
2		cc4f.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		rabexg 4173	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
5	4	ralrimivw 2568	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
6		cc4f.m	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)
7		rabn0m 3475	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∃𝑥 ∈ 𝐴 𝜓)
8	7	ralbii 2500	. . . 4 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)
9	6, 8	sylibr 134	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
10		cc4f.2	. . 3 ⊢ (𝜑 → 𝑁 ≈ ω)
11	1, 5, 9, 10	cc3 7330	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
12		simprl 529	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → 𝑓 Fn 𝑁)
13		elrabi 2914	. . . . . . . 8 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → (𝑓‘𝑛) ∈ 𝐴)
14	13	ralimi 2557	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐴)
15	14	ad2antll 491	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐴)
16		nfcv 2336	. . . . . . 7 ⊢ Ⅎ𝑛𝑁
17		cc4f.a	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
18		nfcv 2336	. . . . . . 7 ⊢ Ⅎ𝑛𝑓
19	16, 17, 18	ffnfvf 5718	. . . . . 6 ⊢ (𝑓:𝑁⟶𝐴 ↔ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐴))
20	12, 15, 19	sylanbrc 417	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → 𝑓:𝑁⟶𝐴)
21		cc4f.3	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))
22	21	elrab 2917	. . . . . . . 8 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜒))
23	22	simprbi 275	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝜒)
24	23	ralimi 2557	. . . . . 6 ⊢ (∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → ∀𝑛 ∈ 𝑁 𝜒)
25	24	ad2antll 491	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → ∀𝑛 ∈ 𝑁 𝜒)
26	20, 25	jca 306	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → (𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
27	26	ex 115	. . 3 ⊢ (𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) → (𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒)))
28	27	eximdv 1891	. 2 ⊢ (𝜑 → (∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒)))
29	11, 28	mpd 13	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Ⅎwnfc 2323  ∀wral 2472  ∃wrex 2473  {crab 2476  Vcvv 2760   class class class wbr 4030  ωcom 4623   Fn wfn 5250  ⟶wf 5251  ‘cfv 5255   ≈ cen 6794  CCHOICEwacc 7324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-2nd 6196  df-er 6589  df-en 6797  df-cc 7325

This theorem is referenced by:  cc4  7332