Theorem cc1 7578

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 531	. . . . . 6 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → 𝑥 ≈ ω)
3		simprr 533	. . . . . . 7 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)
4		elequ2 2208	. . . . . . . . 9 ⊢ (𝑧 = 𝑎 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑎))
5	4	exbidv 1874	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝑎))
6	5	cbvralvw 2781	. . . . . . 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 7577	. . . . 5 ⊢ ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎))
9		exsimpr 1667	. . . . 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 5669	. . . . . . 7 ⊢ (𝑎 = 𝑧 → (𝑓‘𝑎) = (𝑓‘𝑧))
12		id 19	. . . . . . 7 ⊢ (𝑎 = 𝑧 → 𝑎 = 𝑧)
13	11, 12	eleq12d 2303	. . . . . 6 ⊢ (𝑎 = 𝑧 → ((𝑓‘𝑎) ∈ 𝑎 ↔ (𝑓‘𝑧) ∈ 𝑧))
14	13	cbvralvw 2781	. . . . 5 ⊢ (∀𝑎 ∈ 𝑥 (𝑓‘𝑎) ∈ 𝑎 ↔ ∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧)
15	14	exbii 1654	. . . 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 1923	1 ⊢ (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396  ∃wex 1541   ∈ wcel 2203  ∀wral 2520   class class class wbr 4108  ωcom 4711   Fn wfn 5346  ‘cfv 5351   ≈ cen 6972  CCHOICEwacc 7575

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-en 6975  df-cc 7576

This theorem is referenced by: (None)