Theorem isref 22897

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 34887. 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 22896	. . . 4 ⊢ Rel Ref
2	1	brrelex2i 5694	. . 3 ⊢ (𝐴Ref𝐵 → 𝐵 ∈ V)
3	2	anim2i 617	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴Ref𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
4		simpl 483	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → 𝐴 ∈ 𝐶)
5		simpr 485	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → 𝑌 = 𝑋)
6		isref.2	. . . . . . 7 ⊢ 𝑌 = ∪ 𝐵
7		isref.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐴
8	5, 6, 7	3eqtr3g 2794	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐵 = ∪ 𝐴)
9		uniexg 7682	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ∪ 𝐴 ∈ V)
10	9	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐴 ∈ V)
11	8, 10	eqeltrd 2832	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → ∪ 𝐵 ∈ V)
12		uniexb 7703	. . . . 5 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
13	11, 12	sylibr 233	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝑌 = 𝑋) → 𝐵 ∈ V)
14	13	adantrr 715	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → 𝐵 ∈ V)
15	4, 14	jca 512	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
16		unieq 4881	. . . . . 6 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16, 7	eqtr4di 2789	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
18	17	eqeq2d 2742	. . . 4 ⊢ (𝑎 = 𝐴 → (∪ 𝑏 = ∪ 𝑎 ↔ ∪ 𝑏 = 𝑋))
19		raleq 3307	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦))
20	18, 19	anbi12d 631	. . 3 ⊢ (𝑎 = 𝐴 → ((∪ 𝑏 = ∪ 𝑎 ∧ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦) ↔ (∪ 𝑏 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦)))
21		unieq 4881	. . . . . 6 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = ∪ 𝐵)
22	21, 6	eqtr4di 2789	. . . . 5 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = 𝑌)
23	22	eqeq1d 2733	. . . 4 ⊢ (𝑏 = 𝐵 → (∪ 𝑏 = 𝑋 ↔ 𝑌 = 𝑋))
24		rexeq 3308	. . . . 5 ⊢ (𝑏 = 𝐵 → (∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
25	24	ralbidv 3170	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
26	23, 25	anbi12d 631	. . 3 ⊢ (𝑏 = 𝐵 → ((∪ 𝑏 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦) ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
27		df-ref 22893	. . 3 ⊢ Ref = {⟨𝑎, 𝑏⟩ ∣ (∪ 𝑏 = ∪ 𝑎 ∧ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑥 ⊆ 𝑦)}
28	20, 26, 27	brabg 5501	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
29	3, 15, 28	pm5.21nd 800	1 ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  Vcvv 3446   ⊆ wss 3913  ∪ cuni 4870   class class class wbr 5110  Refcref 22890

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-ref 22893

This theorem is referenced by:  refbas  22898  refssex  22899  ssref  22900  refref  22901  reftr  22902  refun0  22903  dissnref  22916  reff  32509  locfinreflem  32510  cmpcref  32520  fnessref  34905  refssfne  34906