Theorem isref 23677

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 36878. 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 23676	. . . 4 ⊢ Rel Ref
2	1	brrelex2i 5717	. . 3 ⊢ (𝐴Ref𝐵 → 𝐵 ∈ V)
3	2	anim2i 628	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴Ref𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
4		simpl 487	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → 𝐴 ∈ 𝐶)
5		simpr 489	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
6		isref.2	. . . . . . 7 ⊢ 𝑌 = ∪ 𝐵
7		isref.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐴
8	5, 6, 7	3eqtr3g 2820	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐵 = ∪ 𝐴)
9		uniexg 7740	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ∪ 𝐴 ∈ V)
10	9	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐴 ∈ V)
11	8, 10	eqeltrd 2862	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐵 ∈ V)
12		uniexb 7761	. . . . 5 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
13	11, 12	sylibr 237	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → 𝐵 ∈ V)
14	13	adantrr 729	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → 𝐵 ∈ V)
15	4, 14	jca 520	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
16		unieq 4882	. . . . . 6 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16, 7	eqtr4di 2815	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
18	17	eqeq2d 2773	. . . 4 ⊢ (𝑎 = 𝐴 → (∪ 𝑏 = ∪ 𝑎 ↔ ∪ 𝑏 = 𝑋))
19		raleq 3319	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦))
20	18, 19	anbi12d 643	. . 3 ⊢ (𝑎 = 𝐴 → ((∪ 𝑏 = ∪ 𝑎 ∧ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦) ↔ (∪ 𝑏 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦)))
21		unieq 4882	. . . . . 6 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = ∪ 𝐵)
22	21, 6	eqtr4di 2815	. . . . 5 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = 𝑌)
23	22	eqeq1d 2764	. . . 4 ⊢ (𝑏 = 𝐵 → (∪ 𝑏 = 𝑋 ↔ 𝑌 = 𝑋))
24		rexeq 3318	. . . . 5 ⊢ (𝑏 = 𝐵 → (∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
25	24	ralbidv 3187	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
26	23, 25	anbi12d 643	. . 3 ⊢ (𝑏 = 𝐵 → ((∪ 𝑏 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦) ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
27		df-ref 23673	. . 3 ⊢ Ref = {⟨𝑎, 𝑏⟩ ∣ (∪ 𝑏 = ∪ 𝑎 ∧ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦)}
28	20, 26, 27	brabg 5523	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
29	3, 15, 28	pm5.21nd 813	1 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ⊆ wss 3904  ∪ cuni 4871   class class class wbr 5108  Refcref 23670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-ref 23673

This theorem is used by:  refbas  23678  refssex  23679  ssref  23680  refref  23681  reftr  23682  refun0  23683  dissnref  23696  reff  34238  locfinreflem  34239  cmpcref  34249  fnessref  36896  refssfne  36897