Theorem cc4f 7463

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 4227	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
5	4	ralrimivw 2604	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
6		cc4f.m	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)
7		rabn0m 3519	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∃𝑥 ∈ 𝐴 𝜓)
8	7	ralbii 2536	. . . 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 7462	. 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 2956	. . . . . . . 8 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → (𝑓‘𝑛) ∈ 𝐴)
14	13	ralimi 2593	. . . . . . 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 2372	. . . . . . 7 ⊢ Ⅎ𝑛𝑁
17		cc4f.a	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
18		nfcv 2372	. . . . . . 7 ⊢ Ⅎ𝑛𝑓
19	16, 17, 18	ffnfvf 5796	. . . . . 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 2959	. . . . . . . 8 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜒))
23	22	simprbi 275	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝜒)
24	23	ralimi 2593	. . . . . 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 1926	. 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 1395  ∃wex 1538   ∈ wcel 2200  Ⅎwnfc 2359  ∀wral 2508  ∃wrex 2509  {crab 2512  Vcvv 2799   class class class wbr 4083  ωcom 4682   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318   ≈ cen 6893  CCHOICEwacc 7456

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-2nd 6293  df-er 6688  df-en 6896  df-cc 7457

This theorem is referenced by:  cc4  7464