Theorem isref 15488

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 15480. On the other hand, the two concepts do seem to have a dual relationship.

Hypotheses

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

Assertion

Ref	Expression
isref	|- (B e. C -> (ARefB <-> (X = Y /\ A.x e. B E.y e. A x C_ y)))

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

Proof of Theorem isref

Step	Hyp	Ref	Expression
1		refrel 15487	. . . . 5 |- Rel Ref
2	1	brrelexi 4029	. . . 4 |- (ARefB -> A e. _V)
3	2	adantl 424	. . 3 |- ((B e. C /\ ARefB) -> A e. _V)
4		simpl 346	. . 3 |- ((B e. C /\ ARefB) -> B e. C)
5	3, 4	jca 310	. 2 |- ((B e. C /\ ARefB) -> (A e. _V /\ B e. C))
6		simpr 350	. . . . . . 7 |- ((B e. C /\ X = Y) -> X = Y)
7		isref.1	. . . . . . 7 |- X = U.A
8		isref.2	. . . . . . 7 |- Y = U.B
9	6, 7, 8	3eqtr3g 1952	. . . . . 6 |- ((B e. C /\ X = Y) -> U.A = U.B)
10		uniexg 3795	. . . . . . 7 |- (B e. C -> U.B e. _V)
11	10	adantr 425	. . . . . 6 |- ((B e. C /\ X = Y) -> U.B e. _V)
12	9, 11	eqeltrd 1971	. . . . 5 |- ((B e. C /\ X = Y) -> U.A e. _V)
13		uniexb 3851	. . . . 5 |- (A e. _V <-> U.A e. _V)
14	12, 13	sylibr 217	. . . 4 |- ((B e. C /\ X = Y) -> A e. _V)
15	14	adantrr 431	. . 3 |- ((B e. C /\ (X = Y /\ A.x e. B E.y e. A x C_ y)) -> A e. _V)
16		simpl 346	. . 3 |- ((B e. C /\ (X = Y /\ A.x e. B E.y e. A x C_ y)) -> B e. C)
17	15, 16	jca 310	. 2 |- ((B e. C /\ (X = Y /\ A.x e. B E.y e. A x C_ y)) -> (A e. _V /\ B e. C))
18		unieq 3185	. . . . . 6 |- (a = A -> U.a = U.A)
19	18, 7	syl6eqr 1946	. . . . 5 |- (a = A -> U.a = X)
20	19	eqeq1d 1892	. . . 4 |- (a = A -> (U.a = U.b <-> X = U.b))
21		rexeq 2267	. . . . 5 |- (a = A -> (E.y e. a x C_ y <-> E.y e. A x C_ y))
22	21	ralbidv 2123	. . . 4 |- (a = A -> (A.x e. b E.y e. a x C_ y <-> A.x e. b E.y e. A x C_ y))
23	20, 22	anbi12d 690	. . 3 |- (a = A -> ((U.a = U.b /\ A.x e. b E.y e. a x C_ y) <-> (X = U.b /\ A.x e. b E.y e. A x C_ y)))
24		unieq 3185	. . . . . 6 |- (b = B -> U.b = U.B)
25	24, 8	syl6eqr 1946	. . . . 5 |- (b = B -> U.b = Y)
26	25	eqeq2d 1895	. . . 4 |- (b = B -> (X = U.b <-> X = Y))
27		raleq 2266	. . . 4 |- (b = B -> (A.x e. b E.y e. A x C_ y <-> A.x e. B E.y e. A x C_ y))
28	26, 27	anbi12d 690	. . 3 |- (b = B -> ((X = U.b /\ A.x e. b E.y e. A x C_ y) <-> (X = Y /\ A.x e. B E.y e. A x C_ y)))
29		df-ref 15464	. . 3 |- Ref = {<.a, b>. | (U.a = U.b /\ A.x e. b E.y e. a x C_ y)}
30	23, 28, 29	brabg 3568	. 2 |- ((A e. _V /\ B e. C) -> (ARefB <-> (X = Y /\ A.x e. B E.y e. A x C_ y)))
31	5, 17, 30	pm5.21nd 744	1 |- (B e. C -> (ARefB <-> (X = Y /\ A.x e. B E.y e. A x C_ y)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  U.cuni 3177   class class class wbr 3338  Refcref 15458

This theorem is referenced by:  refbas 15489  refssex 15490  ssref 15492  refref 15494  reftr 15497  fnessref 15503  refssfne 15504

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-ref 15464

Copyright terms: Public domain