Theorem isfne 36533

Description: The predicate "𝐵 is finer than 𝐴". This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)

Hypotheses

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

Assertion

Ref	Expression
isfne	⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))))

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

Allowed substitution hints:   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem isfne

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnerel 36532	. . . . 5 ⊢ Rel Fne
2	1	brrelex1i 5680	. . . 4 ⊢ (𝐴Fne𝐵 → 𝐴 ∈ V)
3	2	anim1i 615	. . 3 ⊢ ((𝐴Fne𝐵 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐶))
4	3	ancoms 458	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴Fne𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐶))
5		simpr 484	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
6		isfne.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐴
7		isfne.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐵
8	5, 6, 7	3eqtr3g 2794	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝑋 = 𝑌) → ∪ 𝐴 = ∪ 𝐵)
9		simpr 484	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐶 ∧ ∪ 𝐴 = ∪ 𝐵) → ∪ 𝐴 = ∪ 𝐵)
10		uniexg 7685	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐶 → ∪ 𝐵 ∈ V)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐶 ∧ ∪ 𝐴 = ∪ 𝐵) → ∪ 𝐵 ∈ V)
12	9, 11	eqeltrd 2836	. . . . . 6 ⊢ ((𝐵 ∈ 𝐶 ∧ ∪ 𝐴 = ∪ 𝐵) → ∪ 𝐴 ∈ V)
13		uniexb 7709	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
14	12, 13	sylibr 234	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ ∪ 𝐴 = ∪ 𝐵) → 𝐴 ∈ V)
15		simpl 482	. . . . 5 ⊢ ((𝐵 ∈ 𝐶 ∧ ∪ 𝐴 = ∪ 𝐵) → 𝐵 ∈ 𝐶)
16	14, 15	jca 511	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ ∪ 𝐴 = ∪ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐶))
17	8, 16	syldan 591	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝑋 = 𝑌) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐶))
18	17	adantrr 717	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐶))
19		unieq 4874	. . . . . 6 ⊢ (𝑟 = 𝐴 → ∪ 𝑟 = ∪ 𝐴)
20	19, 6	eqtr4di 2789	. . . . 5 ⊢ (𝑟 = 𝐴 → ∪ 𝑟 = 𝑋)
21	20	eqeq1d 2738	. . . 4 ⊢ (𝑟 = 𝐴 → (∪ 𝑟 = ∪ 𝑠 ↔ 𝑋 = ∪ 𝑠))
22		raleq 3293	. . . 4 ⊢ (𝑟 = 𝐴 → (∀𝑥 ∈ 𝑟 𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥)))
23	21, 22	anbi12d 632	. . 3 ⊢ (𝑟 = 𝐴 → ((∪ 𝑟 = ∪ 𝑠 ∧ ∀𝑥 ∈ 𝑟 𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥)) ↔ (𝑋 = ∪ 𝑠 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥))))
24		unieq 4874	. . . . . 6 ⊢ (𝑠 = 𝐵 → ∪ 𝑠 = ∪ 𝐵)
25	24, 7	eqtr4di 2789	. . . . 5 ⊢ (𝑠 = 𝐵 → ∪ 𝑠 = 𝑌)
26	25	eqeq2d 2747	. . . 4 ⊢ (𝑠 = 𝐵 → (𝑋 = ∪ 𝑠 ↔ 𝑋 = 𝑌))
27		ineq1 4165	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
28	27	unieqd 4876	. . . . . 6 ⊢ (𝑠 = 𝐵 → ∪ (𝑠 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥))
29	28	sseq2d 3966	. . . . 5 ⊢ (𝑠 = 𝐵 → (𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
30	29	ralbidv 3159	. . . 4 ⊢ (𝑠 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
31	26, 30	anbi12d 632	. . 3 ⊢ (𝑠 = 𝐵 → ((𝑋 = ∪ 𝑠 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))))
32		df-fne 36531	. . 3 ⊢ Fne = {⟨𝑟, 𝑠⟩ ∣ (∪ 𝑟 = ∪ 𝑠 ∧ ∀𝑥 ∈ 𝑟 𝑥 ⊆ ∪ (𝑠 ∩ 𝒫 𝑥))}
33	23, 31, 32	brabg 5487	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))))
34	4, 18, 33	pm5.21nd 801	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863   class class class wbr 5098  Fnecfne 36530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-fne 36531

This theorem is referenced by:  isfne4  36534