Theorem cc1 7477

Description: Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)

Assertion

Ref	Expression
cc1	⊢ (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))

Distinct variable groups:   𝑤,𝑓,𝑧   𝑥,𝑓,𝑧

Proof of Theorem cc1

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → CCHOICE)
2		simprl 529	. . . . . 6 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → 𝑥 ≈ ω)
3		simprr 531	. . . . . . 7 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)
4		elequ2 2205	. . . . . . . . 9 ⊢ (𝑧 = 𝑎 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑎))
5	4	exbidv 1871	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝑎))
6	5	cbvralvw 2769	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧 ↔ ∀𝑎 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑎)
7	3, 6	sylib 122	. . . . . 6 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → ∀𝑎 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑎)
8	1, 2, 7	ccfunen 7476	. . . . 5 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎))
9		exsimpr 1664	. . . . 5 ⊢ (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎) → ∃𝑓∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎)
10	8, 9	syl 14	. . . 4 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → ∃𝑓∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎)
11		fveq2 5635	. . . . . . 7 ⊢ (𝑎 = 𝑧 → (𝑓‘𝑎) = (𝑓‘𝑧))
12		id 19	. . . . . . 7 ⊢ (𝑎 = 𝑧 → 𝑎 = 𝑧)
13	11, 12	eleq12d 2300	. . . . . 6 ⊢ (𝑎 = 𝑧 → ((𝑓‘𝑎) ∈ 𝑎 ↔ (𝑓‘𝑧) ∈ 𝑧))
14	13	cbvralvw 2769	. . . . 5 ⊢ (∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎 ↔ ∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧)
15	14	exbii 1651	. . . 4 ⊢ (∃𝑓∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎 ↔ ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧)
16	10, 15	sylib 122	. . 3 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧)
17	16	ex 115	. 2 ⊢ (CCHOICE → ((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))
18	17	alrimiv 1920	1 ⊢ (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1393  ∃wex 1538   ∈ wcel 2200  ∀wral 2508   class class class wbr 4086  ωcom 4686   Fn wfn 5319  ‘cfv 5324   ≈ cen 6902  CCHOICEwacc 7474

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-en 6905  df-cc 7475

This theorem is referenced by: (None)