Theorem isfne 32334

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 = U. A
isfne.2	|- Y = U. B

Assertion

Ref	Expression
isfne	|- ( B e. C -> ( A Fne B <-> ( X = Y /\ A. x e. A x C_ U. ( B i^i ~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 32333	. . . . 5 |- Rel Fne
2	1	brrelexi 5158	. . . 4 |- ( A Fne B -> A e. _V )
3	2	anim1i 592	. . 3 |- ( ( A Fne B /\ B e. C ) -> ( A e. _V /\ B e. C ) )
4	3	ancoms 469	. 2 |- ( ( B e. C /\ A Fne B ) -> ( A e. _V /\ B e. C ) )
5		simpr 477	. . . . 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 2679	. . . 4 |- ( ( B e. C /\ X = Y ) -> U. A = U. B )
9		simpr 477	. . . . . . 7 |- ( ( B e. C /\ U. A = U. B ) -> U. A = U. B )
10		uniexg 6955	. . . . . . . 8 |- ( B e. C -> U. B e. _V )
11	10	adantr 481	. . . . . . 7 |- ( ( B e. C /\ U. A = U. B ) -> U. B e. _V )
12	9, 11	eqeltrd 2701	. . . . . 6 |- ( ( B e. C /\ U. A = U. B ) -> U. A e. _V )
13		uniexb 6973	. . . . . 6 |- ( A e. _V <-> U. A e. _V )
14	12, 13	sylibr 224	. . . . 5 |- ( ( B e. C /\ U. A = U. B ) -> A e. _V )
15		simpl 473	. . . . 5 |- ( ( B e. C /\ U. A = U. B ) -> B e. C )
16	14, 15	jca 554	. . . 4 |- ( ( B e. C /\ U. A = U. B ) -> ( A e. _V /\ B e. C ) )
17	8, 16	syldan 487	. . 3 |- ( ( B e. C /\ X = Y ) -> ( A e. _V /\ B e. C ) )
18	17	adantrr 753	. 2 |- ( ( B e. C /\ ( X = Y /\ A. x e. A x C_ U. ( B i^i ~P x ) ) ) -> ( A e. _V /\ B e. C ) )
19		unieq 4444	. . . . . 6 |- ( r = A -> U. r = U. A )
20	19, 6	syl6eqr 2674	. . . . 5 |- ( r = A -> U. r = X )
21	20	eqeq1d 2624	. . . 4 |- ( r = A -> ( U. r = U. s <-> X = U. s ) )
22		raleq 3138	. . . 4 |- ( r = A -> ( A. x e. r x C_ U. ( s i^i ~P x ) <-> A. x e. A x C_ U. ( s i^i ~P x ) ) )
23	21, 22	anbi12d 747	. . 3 |- ( r = A -> ( ( U. r = U. s /\ A. x e. r x C_ U. ( s i^i ~P x ) ) <-> ( X = U. s /\ A. x e. A x C_ U. ( s i^i ~P x ) ) ) )
24		unieq 4444	. . . . . 6 |- ( s = B -> U. s = U. B )
25	24, 7	syl6eqr 2674	. . . . 5 |- ( s = B -> U. s = Y )
26	25	eqeq2d 2632	. . . 4 |- ( s = B -> ( X = U. s <-> X = Y ) )
27		ineq1 3807	. . . . . . 7 |- ( s = B -> ( s i^i ~P x ) = ( B i^i ~P x ) )
28	27	unieqd 4446	. . . . . 6 |- ( s = B -> U. ( s i^i ~P x ) = U. ( B i^i ~P x ) )
29	28	sseq2d 3633	. . . . 5 |- ( s = B -> ( x C_ U. ( s i^i ~P x ) <-> x C_ U. ( B i^i ~P x ) ) )
30	29	ralbidv 2986	. . . 4 |- ( s = B -> ( A. x e. A x C_ U. ( s i^i ~P x ) <-> A. x e. A x C_ U. ( B i^i ~P x ) ) )
31	26, 30	anbi12d 747	. . 3 |- ( s = B -> ( ( X = U. s /\ A. x e. A x C_ U. ( s i^i ~P x ) ) <-> ( X = Y /\ A. x e. A x C_ U. ( B i^i ~P x ) ) ) )
32		df-fne 32332	. . 3 |- Fne = { <. r , s >. | ( U. r = U. s /\ A. x e. r x C_ U. ( s i^i ~P x ) ) }
33	23, 31, 32	brabg 4994	. 2 |- ( ( A e. _V /\ B e. C ) -> ( A Fne B <-> ( X = Y /\ A. x e. A x C_ U. ( B i^i ~P x ) ) ) )
34	4, 18, 33	pm5.21nd 941	1 |- ( B e. C -> ( A Fne B <-> ( X = Y /\ A. x e. A x C_ U. ( B i^i ~P x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-fne 32332

This theorem is referenced by:  isfne4  32335