Theorem reff 33999

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 10401. (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 3945	. . . 4 ⊢ ∪ 𝐵 ⊆ ∪ 𝐵
2		eqid 2737	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
3		eqid 2737	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
4	2, 3	isref 23484	. . . . 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 3965	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 ⊆ ∪ 𝐴)
7	4	simplbda 499	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
8		sseq2 3949	. . . . . 6 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑣 ⊆ 𝑢 ↔ 𝑣 ⊆ (𝑓‘𝑣)))
9	8	ac6sg 10401	. . . . 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 769	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 ⊆ ∪ 𝐴)
14		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴)
15		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑓:𝐴⟶𝐵
16		nfra1 3262	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)
17	15, 16	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
18	14, 17	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))
19		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑣 𝑥 ∈ ∪ 𝐴
20	18, 19	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑣(((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴)
21		simplrl 777	. . . . . . . . . . . . 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	ffvelcdmd 7031	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
24	23	adantlr 716	. . . . . . . . . . 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 778	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
27	26	adantlr 716	. . . . . . . . . . . 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 3227	. . . . . . . . . . . 12 ⊢ ((∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
30	27, 28, 29	syl2anc 585	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
31	30	sselda 3922	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ (𝑓‘𝑣))
32		eleq2 2826	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (𝑓‘𝑣)))
33	32	rspcev 3565	. . . . . . . . . 10 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑥 ∈ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
34	25, 31, 33	syl2anc 585	. . . . . . . . 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 4855	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑥 ∈ 𝑣)
37	35, 36	sylib 218	. . . . . . . . 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 3247	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
39		eluni2 4855	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
40	38, 39	sylibr 234	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → 𝑥 ∈ ∪ 𝐵)
41	13, 40	eqelssd 3944	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 = ∪ 𝐴)
42	26, 22, 29	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
43	8	rspcev 3565	. . . . . . . . 9 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑣 ⊆ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
44	23, 42, 43	syl2anc 585	. . . . . . . 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 3236	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
47	4	ad2antrr 727	. . . . . 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 714	. . . . 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 1935	. . 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 801	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∪ cuni 4851   class class class wbr 5086  ⟶wf 6488  ‘cfv 6492  Refcref 23477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-reg 9500  ax-inf2 9553  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-en 8887  df-r1 9679  df-rank 9680  df-card 9854  df-ac 10029  df-ref 23480

This theorem is referenced by:  locfinreflem  34000