Theorem cc4n 7590

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7589, 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 2827	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	3	ralimi 2607	. . . 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 3538	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∃𝑥 ∈ 𝐴 𝜓)
8	7	ralbii 2550	. . . 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 7587	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
12		simprl 531	. . . . 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 2975	. . . . . . . 8 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜒))
15	14	simprbi 275	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝜒)
16	15	ralimi 2607	. . . . . 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 1929	. 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 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526  Vcvv 2815   class class class wbr 4111  ωcom 4714   Fn wfn 5349  ‘cfv 5354   ≈ cen 6975  CCHOICEwacc 7581

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-2nd 6337  df-er 6769  df-en 6978  df-cc 7582

This theorem is referenced by:  omctfn  13215