Theorem ccfunen 7458

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 6901	. . . . . 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 2350	. . . . . 6 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} = 𝑢
6		vex 2802	. . . . . 6 ⊢ 𝑢 ∈ V
7	5, 6	eqeltri 2302	. . . . 5 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V
8	7	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V)
9	4, 8	opabex3d 6272	. . 3 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∈ V)
10		ccfunen.cc	. . . 4 ⊢ (𝜑 → CCHOICE)
11		df-cc 7457	. . . 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 2205	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑢))
15	14	exbidv 1871	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (∃𝑤 𝑤 ∈ 𝑥 ↔ ∃𝑤 𝑤 ∈ 𝑢))
16	15	cbvralv 2765	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥 ↔ ∀𝑢 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑢)
17		elequ1 2204	. . . . . . . . 9 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑢 ↔ 𝑣 ∈ 𝑢))
18	17	cbvexv 1965	. . . . . . . 8 ⊢ (∃𝑤 𝑤 ∈ 𝑢 ↔ ∃𝑣 𝑣 ∈ 𝑢)
19	18	ralbii 2536	. . . . . . 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 4936	. . . . 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 4105	. . 3 ⊢ (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ≈ ω)
25		dmeq 4923	. . . . . 6 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})
26	25	breq1d 4093	. . . . 5 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (dom 𝑦 ≈ ω ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ≈ ω))
27		sseq2 3248	. . . . . . 7 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
28	25	fneq2d 5412	. . . . . . 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 1871	. . . . 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 2890	. . 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 5412	. . . . . . 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 5767	. . . . . . . . . 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 3225	. . . . . . . 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 2802	. . . . . . . . 9 ⊢ 𝑥 ∈ V
43		vex 2802	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
44	43, 42	fvex 5649	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
45		eleq1 2292	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
46		elequ2 2205	. . . . . . . . . 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 2292	. . . . . . . . . 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 4360	. . . . . . . 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 2603	. . . . 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 1926	. 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 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∀wral 2508  Vcvv 2799   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4083  {copab 4144  ωcom 4682  dom cdm 4719   Fn wfn 5313  ‘cfv 5318   ≈ cen 6893  CCHOICEwacc 7456

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 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

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 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-en 6896  df-cc 7457

This theorem is referenced by:  cc1  7459  cc2lem  7460