Theorem reff 31691

Description: For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 10175. (Contributed by Thierry Arnoux, 12-Jan-2020.)

Assertion

Ref	Expression
reff	⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))

Distinct variable groups:   𝐴,𝑓,𝑣   𝐵,𝑓,𝑣   𝑓,𝑉,𝑣

Proof of Theorem reff

Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3939	. . . 4 ⊢ ∪ 𝐵 ⊆ ∪ 𝐵
2		eqid 2738	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
3		eqid 2738	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
4	2, 3	isref 22568	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)))
5	4	simprbda 498	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 = ∪ 𝐴)
6	1, 5	sseqtrid 3969	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 ⊆ ∪ 𝐴)
7	4	simplbda 499	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
8		sseq2 3943	. . . . . 6 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑣 ⊆ 𝑢 ↔ 𝑣 ⊆ (𝑓‘𝑣)))
9	8	ac6sg 10175	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
10	9	adantr 480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → (∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
11	7, 10	mpd 15	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))
12	6, 11	jca 511	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
13		simplr 765	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 ⊆ ∪ 𝐴)
14		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴)
15		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑓:𝐴⟶𝐵
16		nfra1 3142	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)
17	15, 16	nfan 1903	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
18	14, 17	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))
19		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑣 𝑥 ∈ ∪ 𝐴
20	18, 19	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑣(((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴)
21		simplrl 773	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑓:𝐴⟶𝐵)
22		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
23	21, 22	ffvelrnd 6944	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
24	23	adantlr 711	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
25	24	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → (𝑓‘𝑣) ∈ 𝐵)
26		simplrr 774	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
27	26	adantlr 711	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
28		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
29		rspa 3130	. . . . . . . . . . . 12 ⊢ ((∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
30	27, 28, 29	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
31	30	sselda 3917	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ (𝑓‘𝑣))
32		eleq2 2827	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (𝑓‘𝑣)))
33	32	rspcev 3552	. . . . . . . . . 10 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑥 ∈ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
34	25, 31, 33	syl2anc 583	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
35		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → 𝑥 ∈ ∪ 𝐴)
36		eluni2 4840	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑥 ∈ 𝑣)
37	35, 36	sylib 217	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑣 ∈ 𝐴 𝑥 ∈ 𝑣)
38	20, 34, 37	r19.29af 3259	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
39		eluni2 4840	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
40	38, 39	sylibr 233	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → 𝑥 ∈ ∪ 𝐵)
41	13, 40	eqelssd 3938	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 = ∪ 𝐴)
42	26, 22, 29	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
43	8	rspcev 3552	. . . . . . . . 9 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑣 ⊆ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
44	23, 42, 43	syl2anc 583	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
45	44	ex 412	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → (𝑣 ∈ 𝐴 → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢))
46	18, 45	ralrimi 3139	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
47	4	ad2antrr 722	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → (𝐴Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)))
48	41, 46, 47	mpbir2and 709	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → 𝐴Ref𝐵)
49	48	ex 412	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) → ((𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)) → 𝐴Ref𝐵))
50	49	exlimdv 1937	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) → (∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)) → 𝐴Ref𝐵))
51	50	impr 454	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))) → 𝐴Ref𝐵)
52	12, 51	impbida 797	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∪ cuni 4836   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  Refcref 22561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329  ax-ac2 10150

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-en 8692  df-r1 9453  df-rank 9454  df-card 9628  df-ac 9803  df-ref 22564

This theorem is referenced by:  locfinreflem  31692