Theorem cc4 7643

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 2392	. 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 7642	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀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   class class class wbr 4131  ωcom 4739  ⟶wf 5376  ‘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: (None)