Theorem acfun 7163

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 2739	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
3		abid2 2287	. . . . . 6 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} = 𝑢
4		vex 2729	. . . . . 6 ⊢ 𝑢 ∈ V
5	3, 4	eqeltri 2239	. . . . 5 ⊢ {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V
6	5	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → {𝑣 ∣ 𝑣 ∈ 𝑢} ∈ V)
7	2, 6	opabex3d 6089	. . 3 ⊢ (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∈ V)
8		acfun.ac	. . . 4 ⊢ (𝜑 → CHOICE)
9		df-ac 7162	. . . 4 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
10	8, 9	sylib 121	. . 3 ⊢ (𝜑 → ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
11		sseq2 3166	. . . . . 6 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)}))
12		dmeq 4804	. . . . . . 7 ⊢ (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})
13	12	fneq2d 5279	. . . . . 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 465	. . . . 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 1813	. . . 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 2813	. . 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 522	. . . . . 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 2141	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑢))
21	20	exbidv 1813	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (∃𝑤 𝑤 ∈ 𝑥 ↔ ∃𝑤 𝑤 ∈ 𝑢))
22	21	cbvralv 2692	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥 ↔ ∀𝑢 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑢)
23		elequ1 2140	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑢 ↔ 𝑣 ∈ 𝑢))
24	23	cbvexv 1906	. . . . . . . . . . . 12 ⊢ (∃𝑤 𝑤 ∈ 𝑢 ↔ ∃𝑣 𝑣 ∈ 𝑢)
25	24	ralbii 2472	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑢 ↔ ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
26	22, 25	bitri 183	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥 ↔ ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
27	19, 26	sylib 121	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢)
28		dmopab3 4817	. . . . . . . . 9 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑣 𝑣 ∈ 𝑢 ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} = 𝐴)
29	27, 28	sylib 121	. . . . . . . 8 ⊢ (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} = 𝐴)
30	29	fneq2d 5279	. . . . . . 7 ⊢ (𝜑 → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ↔ 𝑓 Fn 𝐴))
31	30	adantr 274	. . . . . 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 146	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)})) → 𝑓 Fn 𝐴)
33		simplrl 525	. . . . . . . . 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 5615	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, (𝑓‘𝑥)⟩ ∈ 𝑓)
35	32, 34	sylan 281	. . . . . . . . 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 3143	. . . . . . . 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 2729	. . . . . . . . 9 ⊢ 𝑥 ∈ V
38		vex 2729	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
39	38, 37	fvex 5506	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
40		eleq1 2229	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
41		elequ2 2141	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑣 ∈ 𝑢 ↔ 𝑣 ∈ 𝑥))
42	40, 41	anbi12d 465	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢) ↔ (𝑥 ∈ 𝐴 ∧ 𝑣 ∈ 𝑥)))
43		eleq1 2229	. . . . . . . . . 10 ⊢ (𝑣 = (𝑓‘𝑥) → (𝑣 ∈ 𝑥 ↔ (𝑓‘𝑥) ∈ 𝑥))
44	43	anbi2d 460	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑣 ∈ 𝑥) ↔ (𝑥 ∈ 𝐴 ∧ (𝑓‘𝑥) ∈ 𝑥)))
45	37, 39, 42, 44	opelopab 4249	. . . . . . . 8 ⊢ (⟨𝑥, (𝑓‘𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝑢)} ↔ (𝑥 ∈ 𝐴 ∧ (𝑓‘𝑥) ∈ 𝑥))
46	36, 45	sylib 121	. . . . . . 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 113	. . . . . 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 2539	. . . . 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 304	. . . 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 114	. . 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 1868	. 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 103   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  ∀wral 2444  Vcvv 2726   ⊆ wss 3116  ⟨cop 3579  {copab 4042  dom cdm 4604   Fn wfn 5183  ‘cfv 5188  CHOICEwac 7161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ac 7162

This theorem is referenced by:  exmidaclem  7164