Theorem isref 21821

Description: The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 33205. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Hypotheses

Ref	Expression
isref.1	⊢ 𝑋 = ∪ 𝐴
isref.2	⊢ 𝑌 = ∪ 𝐵

Assertion

Ref	Expression
isref	⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵

Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem isref

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		refrel 21820	. . . 4 ⊢ Rel Ref
2	1	brrelex2i 5459	. . 3 ⊢ (𝐴Ref𝐵 → 𝐵 ∈ V)
3	2	anim2i 607	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴Ref𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
4		simpl 475	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → 𝐴 ∈ 𝐶)
5		simpr 477	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
6		isref.2	. . . . . . 7 ⊢ 𝑌 = ∪ 𝐵
7		isref.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐴
8	5, 6, 7	3eqtr3g 2838	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐵 = ∪ 𝐴)
9		uniexg 7285	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ∪ 𝐴 ∈ V)
10	9	adantr 473	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐴 ∈ V)
11	8, 10	eqeltrd 2867	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐵 ∈ V)
12		uniexb 7303	. . . . 5 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
13	11, 12	sylibr 226	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → 𝐵 ∈ V)
14	13	adantrr 704	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → 𝐵 ∈ V)
15	4, 14	jca 504	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
16		unieq 4720	. . . . . 6 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16, 7	syl6eqr 2833	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
18	17	eqeq2d 2789	. . . 4 ⊢ (𝑎 = 𝐴 → (∪ 𝑏 = ∪ 𝑎 ↔ ∪ 𝑏 = 𝑋))
19		raleq 3346	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦))
20	18, 19	anbi12d 621	. . 3 ⊢ (𝑎 = 𝐴 → ((∪ 𝑏 = ∪ 𝑎 ∧ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦) ↔ (∪ 𝑏 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦)))
21		unieq 4720	. . . . . 6 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = ∪ 𝐵)
22	21, 6	syl6eqr 2833	. . . . 5 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = 𝑌)
23	22	eqeq1d 2781	. . . 4 ⊢ (𝑏 = 𝐵 → (∪ 𝑏 = 𝑋 ↔ 𝑌 = 𝑋))
24		rexeq 3347	. . . . 5 ⊢ (𝑏 = 𝐵 → (∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
25	24	ralbidv 3148	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
26	23, 25	anbi12d 621	. . 3 ⊢ (𝑏 = 𝐵 → ((∪ 𝑏 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦) ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
27		df-ref 21817	. . 3 ⊢ Ref = {⟨𝑎, 𝑏⟩ ∣ (∪ 𝑏 = ∪ 𝑎 ∧ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦)}
28	20, 26, 27	brabg 5280	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
29	3, 15, 28	pm5.21nd 789	1 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∀wral 3089  ∃wrex 3090  Vcvv 3416   ⊆ wss 3830  ∪ cuni 4712   class class class wbr 4929  Refcref 21814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-ref 21817

This theorem is referenced by:  refbas  21822  refssex  21823  ssref  21824  refref  21825  reftr  21826  refun0  21827  dissnref  21840  reff  30744  locfinreflem  30745  cmpcref  30755  fnessref  33223  refssfne  33224