Theorem reff 31789

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 10244. (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 3943	. . . 4 ⊢ ∪ 𝐵 ⊆ ∪ 𝐵
2		eqid 2738	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
3		eqid 2738	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
4	2, 3	isref 22660	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)))
5	4	simprbda 499	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 = ∪ 𝐴)
6	1, 5	sseqtrid 3973	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 ⊆ ∪ 𝐴)
7	4	simplbda 500	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
8		sseq2 3947	. . . . . 6 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑣 ⊆ 𝑢 ↔ 𝑣 ⊆ (𝑓‘𝑣)))
9	8	ac6sg 10244	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
10	9	adantr 481	. . . 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 512	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
13		simplr 766	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 ⊆ ∪ 𝐴)
14		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴)
15		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑓:𝐴⟶𝐵
16		nfra1 3144	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)
17	15, 16	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
18	14, 17	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))
19		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑣 𝑥 ∈ ∪ 𝐴
20	18, 19	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑣(((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴)
21		simplrl 774	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑓:𝐴⟶𝐵)
22		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
23	21, 22	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
24	23	adantlr 712	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
25	24	adantr 481	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → (𝑓‘𝑣) ∈ 𝐵)
26		simplrr 775	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
27	26	adantlr 712	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
28		simpr 485	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
29		rspa 3132	. . . . . . . . . . . 12 ⊢ ((∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
30	27, 28, 29	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
31	30	sselda 3921	. . . . . . . . . 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 3561	. . . . . . . . . 10 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑥 ∈ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
34	25, 31, 33	syl2anc 584	. . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → 𝑥 ∈ ∪ 𝐴)
36		eluni2 4843	. . . . . . . . . 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 3262	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
39		eluni2 4843	. . . . . . . 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 3942	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 = ∪ 𝐴)
42	26, 22, 29	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
43	8	rspcev 3561	. . . . . . . . 9 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑣 ⊆ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
44	23, 42, 43	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
45	44	ex 413	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → (𝑣 ∈ 𝐴 → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢))
46	18, 45	ralrimi 3141	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
47	4	ad2antrr 723	. . . . . 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 710	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → 𝐴Ref𝐵)
49	48	ex 413	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) → ((𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)) → 𝐴Ref𝐵))
50	49	exlimdv 1936	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) → (∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)) → 𝐴Ref𝐵))
51	50	impr 455	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))) → 𝐴Ref𝐵)
52	12, 51	impbida 798	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  ⟶wf 6429  ‘cfv 6433  Refcref 22653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399  ax-ac2 10219

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-en 8734  df-r1 9522  df-rank 9523  df-card 9697  df-ac 9872  df-ref 22656

This theorem is referenced by:  locfinreflem  31790