Theorem reff 34170

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 10468. (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 3967	. . . 4 ⊢ ∪ 𝐵 ⊆ ∪ 𝐵
2		eqid 2769	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝐴
3		eqid 2769	. . . . . 6 ⊢ ∪ 𝐵 = ∪ 𝐵
4	2, 3	isref 23631	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)))
5	4	simprbda 503	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 = ∪ 𝐴)
6	1, 5	sseqtrid 3987	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∪ 𝐵 ⊆ ∪ 𝐴)
7	4	simplbda 504	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
8		sseq2 3971	. . . . . 6 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑣 ⊆ 𝑢 ↔ 𝑣 ⊆ (𝑓‘𝑣)))
9	8	ac6sg 10468	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
10	9	adantr 485	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → (∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
11	7, 10	mpd 16	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))
12	6, 11	jca 520	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴Ref𝐵) → (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))
13		simplr 780	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 ⊆ ∪ 𝐴)
14		nfv 1941	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴)
15		nfv 1941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑓:𝐴⟶𝐵
16		nfra1 3295	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)
17	15, 16	nfan 1926	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
18	14, 17	nfan 1926	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))
19		nfv 1941	. . . . . . . . . 10 ⊢ Ⅎ𝑣 𝑥 ∈ ∪ 𝐴
20	18, 19	nfan 1926	. . . . . . . . 9 ⊢ Ⅎ𝑣(((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴)
21		simplrl 788	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑓:𝐴⟶𝐵)
22		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
23	21, 22	ffvelcdmd 7078	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
24	23	adantlr 727	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → (𝑓‘𝑣) ∈ 𝐵)
25	24	adantr 485	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → (𝑓‘𝑣) ∈ 𝐵)
26		simplrr 789	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
27	26	adantlr 727	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))
28		simpr 489	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴)
29		rspa 3260	. . . . . . . . . . . 12 ⊢ ((∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
30	27, 28, 29	syl2anc 595	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
31	30	sselda 3945	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ (𝑓‘𝑣))
32		eleq2 2858	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑓‘𝑣) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (𝑓‘𝑣)))
33	32	rspcev 3590	. . . . . . . . . 10 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑥 ∈ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
34	25, 31, 33	syl2anc 595	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ 𝑣) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
35		eluni2 4877	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑥 ∈ 𝑣)
36	35	bilani 509	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑣 ∈ 𝐴 𝑥 ∈ 𝑣)
37	20, 34, 36	r19.29af 3280	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
38		eluni2 4877	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑢 ∈ 𝐵 𝑥 ∈ 𝑢)
39	37, 38	sylibr 237	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑥 ∈ ∪ 𝐴) → 𝑥 ∈ ∪ 𝐵)
40	13, 39	eqelssd 3966	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∪ 𝐵 = ∪ 𝐴)
41	26, 22, 29	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → 𝑣 ⊆ (𝑓‘𝑣))
42	8	rspcev 3590	. . . . . . . . 9 ⊢ (((𝑓‘𝑣) ∈ 𝐵 ∧ 𝑣 ⊆ (𝑓‘𝑣)) → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
43	23, 41, 42	syl2anc 595	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) ∧ 𝑣 ∈ 𝐴) → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
44	43	ex 417	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → (𝑣 ∈ 𝐴 → ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢))
45	18, 44	ralrimi 3269	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)
46	4	ad2antrr 738	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → (𝐴Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝐵 𝑣 ⊆ 𝑢)))
47	40, 45, 46	mpbir2and 725	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) ∧ (𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))) → 𝐴Ref𝐵)
48	47	ex 417	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) → ((𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)) → 𝐴Ref𝐵))
49	48	exlimdv 1960	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝐵 ⊆ ∪ 𝐴) → (∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)) → 𝐴Ref𝐵))
50	49	impr 459	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))) → 𝐴Ref𝐵)
51	12, 50	impbida 812	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ⊆ wss 3913  ∪ cuni 4873   class class class wbr 5110  ⟶wf 6529  ‘cfv 6533  Refcref 23624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-reg 9550  ax-inf2 9606  ax-ac2 10443

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-en 8940  df-r1 9732  df-rank 9733  df-card 9921  df-ac 10096  df-ref 23627

This theorem is referenced by:  locfinreflem  34171