Theorem cc1 7527

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 2207	. . . . . . . . 9 ⊢ (𝑧 = 𝑎 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑎))
5	4	exbidv 1873	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝑎))
6	5	cbvralvw 2772	. . . . . . 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 7526	. . . . 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 5648	. . . . . . 7 ⊢ (𝑎 = 𝑧 → (𝑓‘𝑎) = (𝑓‘𝑧))
12		id 19	. . . . . . 7 ⊢ (𝑎 = 𝑧 → 𝑎 = 𝑧)
13	11, 12	eleq12d 2302	. . . . . 6 ⊢ (𝑎 = 𝑧 → ((𝑓‘𝑎) ∈ 𝑎 ↔ (𝑓‘𝑧) ∈ 𝑧))
14	13	cbvralvw 2772	. . . . 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 1922	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 2202  ∀wral 2511   class class class wbr 4093  ωcom 4694   Fn wfn 5328  ‘cfv 5333   ≈ cen 6950  CCHOICEwacc 7524

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-en 6953  df-cc 7525

This theorem is referenced by: (None)