Theorem cc4n 7333

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7332, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ ω. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)

Hypotheses

Ref	Expression
cc4n.cc	⊢ (𝜑 → CCHOICE)
cc4n.1	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉)
cc4n.2	⊢ (𝜑 → 𝑁 ≈ ω)
cc4n.3	⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))
cc4n.m	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
cc4n	⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀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 cc4n

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cc4n.cc	. . 3 ⊢ (𝜑 → CCHOICE)
2		cc4n.1	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉)
3		elex 2771	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	3	ralimi 2557	. . . 4 ⊢ (∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
5	2, 4	syl 14	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
6		cc4n.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		cc4n.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	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → 𝑓 Fn 𝑁)
13		cc4n.3	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))
14	13	elrab 2917	. . . . . . . 8 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜒))
15	14	simprbi 275	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝜒)
16	15	ralimi 2557	. . . . . 6 ⊢ (∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → ∀𝑛 ∈ 𝑁 𝜒)
17	16	ad2antll 491	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → ∀𝑛 ∈ 𝑁 𝜒)
18	12, 17	jca 306	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) → (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒))
19	18	ex 115	. . 3 ⊢ (𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) → (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒)))
20	19	eximdv 1891	. 2 ⊢ (𝜑 → (∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒)))
21	11, 20	mpd 13	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  {crab 2476  Vcvv 2760   class class class wbr 4030  ωcom 4623   Fn wfn 5250  ‘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:  omctfn  12603