Theorem isfne 15480

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.

Hypotheses

Ref	Expression
isfne.1	|- X = U.A
isfne.2	|- Y = U.B

Assertion

Ref	Expression
isfne	|- (B e. C -> (AFneB <-> (X = Y /\ A.x e. A x C_ U.(B i^i ~Px))))

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

Proof of Theorem isfne

Step	Hyp	Ref	Expression
1		fnerel 15479	. . . . 5 |- Rel Fne
2	1	brrelexi 4029	. . . 4 |- (AFneB -> A e. _V)
3	2	anim1i 361	. . 3 |- ((AFneB /\ B e. C) -> (A e. _V /\ B e. C))
4	3	ancoms 484	. 2 |- ((B e. C /\ AFneB) -> (A e. _V /\ B e. C))
5		simpr 350	. . . . 5 |- ((B e. C /\ X = Y) -> X = Y)
6		isfne.1	. . . . 5 |- X = U.A
7		isfne.2	. . . . 5 |- Y = U.B
8	5, 6, 7	3eqtr3g 1952	. . . 4 |- ((B e. C /\ X = Y) -> U.A = U.B)
9		simpr 350	. . . . . . 7 |- ((B e. C /\ U.A = U.B) -> U.A = U.B)
10		uniexg 3795	. . . . . . . 8 |- (B e. C -> U.B e. _V)
11	10	adantr 425	. . . . . . 7 |- ((B e. C /\ U.A = U.B) -> U.B e. _V)
12	9, 11	eqeltrd 1971	. . . . . 6 |- ((B e. C /\ U.A = U.B) -> U.A e. _V)
13		uniexb 3851	. . . . . 6 |- (A e. _V <-> U.A e. _V)
14	12, 13	sylibr 217	. . . . 5 |- ((B e. C /\ U.A = U.B) -> A e. _V)
15		simpl 346	. . . . 5 |- ((B e. C /\ U.A = U.B) -> B e. C)
16	14, 15	jca 310	. . . 4 |- ((B e. C /\ U.A = U.B) -> (A e. _V /\ B e. C))
17	8, 16	syldan 516	. . 3 |- ((B e. C /\ X = Y) -> (A e. _V /\ B e. C))
18	17	adantrr 431	. 2 |- ((B e. C /\ (X = Y /\ A.x e. A x C_ U.(B i^i ~Px))) -> (A e. _V /\ B e. C))
19		unieq 3185	. . . . . 6 |- (r = A -> U.r = U.A)
20	19, 6	syl6eqr 1946	. . . . 5 |- (r = A -> U.r = X)
21	20	eqeq1d 1892	. . . 4 |- (r = A -> (U.r = U.s <-> X = U.s))
22		raleq 2266	. . . 4 |- (r = A -> (A.x e. r x C_ U.(s i^i ~Px) <-> A.x e. A x C_ U.(s i^i ~Px)))
23	21, 22	anbi12d 690	. . 3 |- (r = A -> ((U.r = U.s /\ A.x e. r x C_ U.(s i^i ~Px)) <-> (X = U.s /\ A.x e. A x C_ U.(s i^i ~Px))))
24		unieq 3185	. . . . . 6 |- (s = B -> U.s = U.B)
25	24, 7	syl6eqr 1946	. . . . 5 |- (s = B -> U.s = Y)
26	25	eqeq2d 1895	. . . 4 |- (s = B -> (X = U.s <-> X = Y))
27		ineq1 2789	. . . . . . 7 |- (s = B -> (s i^i ~Px) = (B i^i ~Px))
28	27	unieqd 3188	. . . . . 6 |- (s = B -> U.(s i^i ~Px) = U.(B i^i ~Px))
29	28	sseq2d 2645	. . . . 5 |- (s = B -> (x C_ U.(s i^i ~Px) <-> x C_ U.(B i^i ~Px)))
30	29	ralbidv 2123	. . . 4 |- (s = B -> (A.x e. A x C_ U.(s i^i ~Px) <-> A.x e. A x C_ U.(B i^i ~Px)))
31	26, 30	anbi12d 690	. . 3 |- (s = B -> ((X = U.s /\ A.x e. A x C_ U.(s i^i ~Px)) <-> (X = Y /\ A.x e. A x C_ U.(B i^i ~Px))))
32		df-fne 15463	. . 3 |- Fne = {<.r, s>. | (U.r = U.s /\ A.x e. r x C_ U.(s i^i ~Px))}
33	23, 31, 32	brabg 3568	. 2 |- ((A e. _V /\ B e. C) -> (AFneB <-> (X = Y /\ A.x e. A x C_ U.(B i^i ~Px))))
34	4, 18, 33	pm5.21nd 744	1 |- (B e. C -> (AFneB <-> (X = Y /\ A.x e. A x C_ U.(B i^i ~Px))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  ~Pcpw 3032  U.cuni 3177   class class class wbr 3338  Fnecfne 15457

This theorem is referenced by:  isfne2 15481  fnebas 15483

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-fne 15463

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