Theorem acfun 7301

Description: A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)

Hypotheses

Ref	Expression
acfun.ac	⊢ (𝜑 → CHOICE)
acfun.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
acfun.m	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)

Assertion

Ref	Expression
acfun	⊢ (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)   𝑉(𝑥,𝑤,𝑓)

Proof of Theorem acfun

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

Step	Hyp	Ref	Expression
1		acfun.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	elexd 2784	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
3		abid2 2325	. . . . . 6 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} = 𝑢
4		vex 2774	. . . . . 6 ⊢ 𝑢 ∈ V
5	3, 4	eqeltri 2277	. . . . 5 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V
6	5	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V)
7	2, 6	opabex3d 6196	. . 3 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∈ V)
8		acfun.ac	. . . 4 ⊢ (𝜑 → CHOICE)
9		df-ac 7300	. . . 4 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
10	8, 9	sylib 122	. . 3 ⊢ (𝜑 → ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
11		sseq2 3216	. . . . . 6 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
12		dmeq 4876	. . . . . . 7 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})
13	12	fneq2d 5359	. . . . . 6 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (𝑓 Fn dom 𝑦 ↔ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
14	11, 13	anbi12d 473	. . . . 5 ⊢ (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)})))
15	14	exbidv 1847	. . . 4 ⊢ (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)})))
16	15	spcgv 2859	. . 3 ⊢ ({⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∈ V → (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})))
17	7, 10, 16	sylc 62	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
18		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)})
19		acfun.m	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)
20		elequ2 2180	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑢))
21	20	exbidv 1847	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (∃𝑤 𝑤 ∈ 𝑥 ↔ ∃𝑤 𝑤 ∈ 𝑢))
22	21	cbvralv 2737	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥 ↔ ∀𝑢 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑢)
23		elequ1 2179	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑢 ↔ 𝑣 ∈ 𝑢))
24	23	cbvexv 1941	. . . . . . . . . . . 12 ⊢ (∃𝑤 𝑤 ∈ 𝑢 ↔ ∃𝑣 𝑣 ∈ 𝑢)
25	24	ralbii 2511	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑢 ↔ ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
26	22, 25	bitri 184	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥 ↔ ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
27	19, 26	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
28		dmopab3 4889	. . . . . . . . 9 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢 ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} = 𝐴)
29	27, 28	sylib 122	. . . . . . . 8 ⊢ (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} = 𝐴)
30	29	fneq2d 5359	. . . . . . 7 ⊢ (𝜑 → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ↔ 𝑓 Fn 𝐴))
31	30	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))
32	18, 31	mpbid 147	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → 𝑓 Fn 𝐴)
33		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)})
34		fnopfv 5704	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝑓‘𝑥)⟩ ∈ 𝑓)
35	32, 34	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)
36	33, 35	sseldd 3193	. . . . . . . 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)})
37		vex 2774	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		vex 2774	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
39	38, 37	fvex 5590	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
40		eleq1 2267	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
41		elequ2 2180	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑣 ∈ 𝑢 ↔ 𝑣 ∈ 𝑥))
42	40, 41	anbi12d 473	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢) ↔ (𝑥 ∈ 𝐴 ∧ 𝑣 ∈ 𝑥)))
43		eleq1 2267	. . . . . . . . . 10 ⊢ (𝑣 = (𝑓‘𝑥) → (𝑣 ∈ 𝑥 ↔ (𝑓‘𝑥) ∈ 𝑥))
44	43	anbi2d 464	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑣 ∈ 𝑥) ↔ (𝑥 ∈ 𝐴 ∧ (𝑓‘𝑥) ∈ 𝑥)))
45	37, 39, 42, 44	opelopab 4316	. . . . . . . 8 ⊢ (⟨𝑥, (𝑓‘𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ↔ (𝑥 ∈ 𝐴 ∧ (𝑓‘𝑥) ∈ 𝑥))
46	36, 45	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))
47	46	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)
48	47	ralrimiva 2578	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)
49	32, 48	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))
50	49	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)))
51	50	eximdv 1902	. 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)))
52	17, 51	mpd 13	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1370   = wceq 1372  ∃wex 1514   ∈ wcel 2175  {cab 2190  ∀wral 2483  Vcvv 2771   ⊆ wss 3165  ⟨cop 3635  {copab 4103  dom cdm 4673   Fn wfn 5263  ‘cfv 5268  CHOICEwac 7299

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-ac 7300

This theorem is referenced by:  exmidaclem  7302