Theorem reff 33785

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 10557. (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 4031	. . . 4 ⊢ ∪ 𝐵 ⊆ ∪ 𝐵
2		eqid 2740	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
3		eqid 2740	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
4	2, 3	isref 23538	. . . . 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 4061	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 ⊆ ∪ 𝐴)
7	4	simplbda 499	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
8		sseq2 4035	. . . . . 6 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑣 ⊆ 𝑢 ↔ 𝑣 ⊆ (𝑓‘𝑣)))
9	8	ac6sg 10557	. . . . 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 768	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 ⊆ ∪ 𝐴)
14		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴)
15		nfv 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑓:𝐴⟶𝐵
16		nfra1 3290	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)
17	15, 16	nfan 1898	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
18	14, 17	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))
19		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑣 𝑥 ∈ ∪ 𝐴
20	18, 19	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑣(((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴)
21		simplrl 776	. . . . . . . . . . . . 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 7119	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
24	23	adantlr 714	. . . . . . . . . . 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 777	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
27	26	adantlr 714	. . . . . . . . . . . 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 3254	. . . . . . . . . . . 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 4008	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ (𝑓‘𝑣))
32		eleq2 2833	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (𝑓‘𝑣)))
33	32	rspcev 3635	. . . . . . . . . 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 4935	. . . . . . . . . 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 3274	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
39		eluni2 4935	. . . . . . . 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 4030	. . . . . 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 3635	. . . . . . . . 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 3263	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
47	4	ad2antrr 725	. . . . . 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 712	. . . . 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 1932	. . 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 800	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∪ cuni 4931   class class class wbr 5166  ⟶wf 6569  ‘cfv 6573  Refcref 23531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-en 9004  df-r1 9833  df-rank 9834  df-card 10008  df-ac 10185  df-ref 23534

This theorem is referenced by:  locfinreflem  33786