Theorem cc4n 7644

Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7643, 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 2833	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	3	ralimi 2613	. . . 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 3549	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∃𝑥 ∈ 𝐴 𝜓)
8	7	ralbii 2556	. . . 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 7641	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
12		simprl 535	. . . . 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 2982	. . . . . . . 8 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜒))
15	14	simprbi 275	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝜒)
16	15	ralimi 2613	. . . . . 6 ⊢ (∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → ∀𝑛 ∈ 𝑁 𝜒)
17	16	ad2antll 495	. . . . 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 1933	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821   class class class wbr 4131  ωcom 4739   Fn wfn 5375  ‘cfv 5380   ≈ cen 7025  CCHOICEwacc 7635

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-2nd 6379  df-er 6811  df-en 7028  df-cc 7636

This theorem is used by:  omctfn  13393