Theorem ccfunen 7383

Description: Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)

Hypotheses

Ref	Expression
ccfunen.cc	⊢ (𝜑 → CCHOICE)
ccfunen.a	⊢ (𝜑 → 𝐴 ≈ ω)
ccfunen.m	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)

Assertion

Ref	Expression
ccfunen	⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Distinct variable groups:   𝐴,𝑓,𝑥   𝜑,𝑓,𝑥   𝑥,𝑤

Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)

Proof of Theorem ccfunen

Dummy variables 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccfunen.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ≈ ω)
2		encv 6840	. . . . . 6 ⊢ (𝐴 ≈ ω → (𝐴 ∈ V ∧ ω ∈ V))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ V ∧ ω ∈ V))
4	3	simpld 112	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
5		abid2 2327	. . . . . 6 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} = 𝑢
6		vex 2776	. . . . . 6 ⊢ 𝑢 ∈ V
7	5, 6	eqeltri 2279	. . . . 5 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V
8	7	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V)
9	4, 8	opabex3d 6213	. . 3 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∈ V)
10		ccfunen.cc	. . . 4 ⊢ (𝜑 → CCHOICE)
11		df-cc 7382	. . . 4 ⊢ (CCHOICE ↔ ∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦)))
12	10, 11	sylib 122	. . 3 ⊢ (𝜑 → ∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦)))
13		ccfunen.m	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)
14		elequ2 2182	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑢))
15	14	exbidv 1849	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (∃𝑤 𝑤 ∈ 𝑥 ↔ ∃𝑤 𝑤 ∈ 𝑢))
16	15	cbvralv 2739	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥 ↔ ∀𝑢 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑢)
17		elequ1 2181	. . . . . . . . 9 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑢 ↔ 𝑣 ∈ 𝑢))
18	17	cbvexv 1943	. . . . . . . 8 ⊢ (∃𝑤 𝑤 ∈ 𝑢 ↔ ∃𝑣 𝑣 ∈ 𝑢)
19	18	ralbii 2513	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑢 ↔ ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
20	16, 19	bitri 184	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥 ↔ ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
21	13, 20	sylib 122	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
22		dmopab3 4896	. . . . 5 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢 ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} = 𝐴)
23	21, 22	sylib 122	. . . 4 ⊢ (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} = 𝐴)
24	23, 1	eqbrtrd 4069	. . 3 ⊢ (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ≈ ω)
25		dmeq 4883	. . . . . 6 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})
26	25	breq1d 4057	. . . . 5 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (dom 𝑦 ≈ ω ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ≈ ω))
27		sseq2 3218	. . . . . . 7 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
28	25	fneq2d 5370	. . . . . . 7 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (𝑓 Fn dom 𝑦 ↔ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
29	27, 28	anbi12d 473	. . . . . 6 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → ((𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})))
30	29	exbidv 1849	. . . . 5 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})))
31	26, 30	imbi12d 234	. . . 4 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → ((dom 𝑦 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦)) ↔ (dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ≈ ω → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))))
32	31	spcgv 2861	. . 3 ⊢ ({⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∈ V → (∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦)) → (dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ≈ ω → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))))
33	9, 12, 24, 32	syl3c 63	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
34		simprr 531	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})
35	23	fneq2d 5370	. . . . . . 7 ⊢ (𝜑 → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ↔ 𝑓 Fn 𝐴))
36	35	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ↔ 𝑓 Fn 𝐴))
37	34, 36	mpbid 147	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → 𝑓 Fn 𝐴)
38		simplrl 535	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) ∧ 𝑥 ∈ 𝐴) → 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})
39		fnopfv 5717	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝑓‘𝑥)⟩ ∈ 𝑓)
40	37, 39	sylan 283	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝑓‘𝑥)⟩ ∈ 𝑓)
41	38, 40	sseldd 3195	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝑓‘𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})
42		vex 2776	. . . . . . . . 9 ⊢ 𝑥 ∈ V
43		vex 2776	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
44	43, 42	fvex 5603	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
45		eleq1 2269	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
46		elequ2 2182	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑣 ∈ 𝑢 ↔ 𝑣 ∈ 𝑥))
47	45, 46	anbi12d 473	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢) ↔ (𝑥 ∈ 𝐴 ∧ 𝑣 ∈ 𝑥)))
48		eleq1 2269	. . . . . . . . . 10 ⊢ (𝑣 = (𝑓‘𝑥) → (𝑣 ∈ 𝑥 ↔ (𝑓‘𝑥) ∈ 𝑥))
49	48	anbi2d 464	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑣 ∈ 𝑥) ↔ (𝑥 ∈ 𝐴 ∧ (𝑓‘𝑥) ∈ 𝑥)))
50	42, 44, 47, 49	opelopab 4322	. . . . . . . 8 ⊢ (⟨𝑥, (𝑓‘𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ↔ (𝑥 ∈ 𝐴 ∧ (𝑓‘𝑥) ∈ 𝑥))
51	41, 50	sylib 122	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ (𝑓‘𝑥) ∈ 𝑥))
52	51	simprd 114	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝑥)
53	52	ralrimiva 2580	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)
54	37, 53	jca 306	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
55	54	ex 115	. . 3 ⊢ (𝜑 → ((𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}) → (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)))
56	55	eximdv 1904	. 2 ⊢ (𝜑 → (∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)))
57	33, 56	mpd 13	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {cab 2192  ∀wral 2485  Vcvv 2773   ⊆ wss 3167  ⟨cop 3637   class class class wbr 4047  {copab 4108  ωcom 4642  dom cdm 4679   Fn wfn 5271  ‘cfv 5276   ≈ cen 6832  CCHOICEwacc 7381

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-en 6835  df-cc 7382

This theorem is referenced by:  cc1  7384  cc2lem  7385