Theorem cc4 7424

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, 1-May-2024.)

Hypotheses

Ref	Expression
cc4.cc	⊢ (𝜑 → CCHOICE)
cc4.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
cc4.2	⊢ (𝜑 → 𝑁 ≈ ω)
cc4.3	⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))
cc4.m	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
cc4	⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))

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

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

Proof of Theorem cc4

Step	Hyp	Ref	Expression
1		cc4.cc	. 2 ⊢ (𝜑 → CCHOICE)
2		cc4.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		nfcv 2352	. 2 ⊢ Ⅎ𝑛𝐴
4		cc4.2	. 2 ⊢ (𝜑 → 𝑁 ≈ ω)
5		cc4.3	. 2 ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))
6		cc4.m	. 2 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)
7	1, 2, 3, 4, 5, 6	cc4f 7423	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1375  ∃wex 1518   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489   class class class wbr 4062  ωcom 4659  ⟶wf 5290  ‘cfv 5294   ≈ cen 6855  CCHOICEwacc 7416

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-2nd 6257  df-er 6650  df-en 6858  df-cc 7417

This theorem is referenced by: (None)