Theorem refssfne 15504

Description: A cover is a refinement iff it is a subcover of something which is both finer and a refinement.

Hypotheses

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

Assertion

Ref	Expression
refssfne	|- ((B e. C /\ X = Y) -> (ARefB <-> E.c(B C_ c /\ A(Fne i^i Ref)c)))

Distinct variable groups:   A,c   B,c   C,c   X,c   Y,c

Proof of Theorem refssfne

Step	Hyp	Ref	Expression
1		refrel 15487	. . . . . . 7 |- Rel Ref
2	1	brrelexi 4029	. . . . . 6 |- (ARefB -> A e. _V)
3	2	3ad2ant3 899	. . . . 5 |- ((B e. C /\ X = Y /\ ARefB) -> A e. _V)
4		simp1 876	. . . . 5 |- ((B e. C /\ X = Y /\ ARefB) -> B e. C)
5		unexg 3798	. . . . 5 |- ((A e. _V /\ B e. C) -> (A u. B) e. _V)
6	3, 4, 5	syl11anc 524	. . . 4 |- ((B e. C /\ X = Y /\ ARefB) -> (A u. B) e. _V)
7		brin 3388	. . . . . 6 |- (A(Fne i^i Ref)(A u. B) <-> (AFne(A u. B) /\ ARef(A u. B)))
8		ssun1 2767	. . . . . . . 8 |- A C_ (A u. B)
9	8	a1i 8	. . . . . . 7 |- ((B e. C /\ X = Y /\ ARefB) -> A C_ (A u. B))
10		uneq2 2749	. . . . . . . . 9 |- (X = Y -> (X u. X) = (X u. Y))
11	10	3ad2ant2 898	. . . . . . . 8 |- ((B e. C /\ X = Y /\ ARefB) -> (X u. X) = (X u. Y))
12		unidm 2743	. . . . . . . 8 |- (X u. X) = X
13		refssfne.1	. . . . . . . . 9 |- X = U.A
14		refssfne.2	. . . . . . . . 9 |- Y = U.B
15	13, 14	uneq12i 2753	. . . . . . . 8 |- (X u. Y) = (U.A u. U.B)
16	11, 12, 15	3eqtr3g 1952	. . . . . . 7 |- ((B e. C /\ X = Y /\ ARefB) -> X = (U.A u. U.B))
17		uniun 3196	. . . . . . . . 9 |- U.(A u. B) = (U.A u. U.B)
18	17	eqcomi 1888	. . . . . . . 8 |- (U.A u. U.B) = U.(A u. B)
19	13, 18	fness 15491	. . . . . . 7 |- (((A u. B) e. _V /\ A C_ (A u. B) /\ X = (U.A u. U.B)) -> AFne(A u. B))
20	6, 9, 16, 19	syl111anc 1100	. . . . . 6 |- ((B e. C /\ X = Y /\ ARefB) -> AFne(A u. B))
21	13, 18	isref 15488	. . . . . . . 8 |- ((A u. B) e. _V -> (ARef(A u. B) <-> (X = (U.A u. U.B) /\ A.x e. (A u. B)E.y e. A x C_ y)))
22	6, 21	syl 12	. . . . . . 7 |- ((B e. C /\ X = Y /\ ARefB) -> (ARef(A u. B) <-> (X = (U.A u. U.B) /\ A.x e. (A u. B)E.y e. A x C_ y)))
23		ssid 2634	. . . . . . . . . . . 12 |- x C_ x
24		sseq2 2639	. . . . . . . . . . . . 13 |- (y = x -> (x C_ y <-> x C_ x))
25	24	rcla4ev 2381	. . . . . . . . . . . 12 |- ((x e. A /\ x C_ x) -> E.y e. A x C_ y)
26	23, 25	mpan2 760	. . . . . . . . . . 11 |- (x e. A -> E.y e. A x C_ y)
27	26	a1i 8	. . . . . . . . . 10 |- ((B e. C /\ X = Y /\ ARefB) -> (x e. A -> E.y e. A x C_ y))
28		refssex 15490	. . . . . . . . . . . 12 |- ((B e. C /\ ARefB /\ x e. B) -> E.y e. A x C_ y)
29	28	3expia 1069	. . . . . . . . . . 11 |- ((B e. C /\ ARefB) -> (x e. B -> E.y e. A x C_ y))
30	29	3adant2 895	. . . . . . . . . 10 |- ((B e. C /\ X = Y /\ ARefB) -> (x e. B -> E.y e. A x C_ y))
31	27, 30	jaod 469	. . . . . . . . 9 |- ((B e. C /\ X = Y /\ ARefB) -> ((x e. A \/ x e. B) -> E.y e. A x C_ y))
32		elun 2741	. . . . . . . . 9 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
33	31, 32	syl5ib 223	. . . . . . . 8 |- ((B e. C /\ X = Y /\ ARefB) -> (x e. (A u. B) -> E.y e. A x C_ y))
34	33	r19.21aiv 2175	. . . . . . 7 |- ((B e. C /\ X = Y /\ ARefB) -> A.x e. (A u. B)E.y e. A x C_ y)
35	22, 16, 34	mpbir2and 802	. . . . . 6 |- ((B e. C /\ X = Y /\ ARefB) -> ARef(A u. B))
36	7, 20, 35	sylanbrc 527	. . . . 5 |- ((B e. C /\ X = Y /\ ARefB) -> A(Fne i^i Ref)(A u. B))
37		ssun2 2768	. . . . 5 |- B C_ (A u. B)
38	36, 37	jctil 316	. . . 4 |- ((B e. C /\ X = Y /\ ARefB) -> (B C_ (A u. B) /\ A(Fne i^i Ref)(A u. B)))
39		sseq2 2639	. . . . . 6 |- (c = (A u. B) -> (B C_ c <-> B C_ (A u. B)))
40		breq2 3342	. . . . . 6 |- (c = (A u. B) -> (A(Fne i^i Ref)c <-> A(Fne i^i Ref)(A u. B)))
41	39, 40	anbi12d 690	. . . . 5 |- (c = (A u. B) -> ((B C_ c /\ A(Fne i^i Ref)c) <-> (B C_ (A u. B) /\ A(Fne i^i Ref)(A u. B))))
42	41	cla4egv 2365	. . . 4 |- ((A u. B) e. _V -> ((B C_ (A u. B) /\ A(Fne i^i Ref)(A u. B)) -> E.c(B C_ c /\ A(Fne i^i Ref)c)))
43	6, 38, 42	sylc 83	. . 3 |- ((B e. C /\ X = Y /\ ARefB) -> E.c(B C_ c /\ A(Fne i^i Ref)c))
44	43	3expia 1069	. 2 |- ((B e. C /\ X = Y) -> (ARefB -> E.c(B C_ c /\ A(Fne i^i Ref)c)))
45		simpll 448	. . . . 5 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> B e. C)
46		brin 3388	. . . . . . 7 |- (A(Fne i^i Ref)c <-> (AFnec /\ ARefc))
47	46	simprbi 353	. . . . . 6 |- (A(Fne i^i Ref)c -> ARefc)
48	47	ad2antll 443	. . . . 5 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> ARefc)
49		simprl 450	. . . . . 6 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> B C_ c)
50		eqcom 1886	. . . . . . . . 9 |- (X = Y <-> Y = X)
51	50	biimpi 168	. . . . . . . 8 |- (X = Y -> Y = X)
52	51	ad2antlr 441	. . . . . . 7 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> Y = X)
53		visset 2295	. . . . . . . . . 10 |- c e. _V
54		eqid 1884	. . . . . . . . . . 11 |- U.c = U.c
55	13, 54	refbas 15489	. . . . . . . . . 10 |- ((c e. _V /\ ARefc) -> X = U.c)
56	53, 55	mpan 759	. . . . . . . . 9 |- (ARefc -> X = U.c)
57	47, 56	syl 12	. . . . . . . 8 |- (A(Fne i^i Ref)c -> X = U.c)
58	57	ad2antll 443	. . . . . . 7 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> X = U.c)
59	52, 58	eqtrd 1925	. . . . . 6 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> Y = U.c)
60	14, 54	ssref 15492	. . . . . 6 |- ((B e. C /\ B C_ c /\ Y = U.c) -> cRefB)
61	45, 49, 59, 60	syl111anc 1100	. . . . 5 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> cRefB)
62		reftr 15497	. . . . 5 |- ((B e. C /\ ARefc /\ cRefB) -> ARefB)
63	45, 48, 61, 62	syl111anc 1100	. . . 4 |- (((B e. C /\ X = Y) /\ (B C_ c /\ A(Fne i^i Ref)c)) -> ARefB)
64	63	ex 402	. . 3 |- ((B e. C /\ X = Y) -> ((B C_ c /\ A(Fne i^i Ref)c) -> ARefB))
65	64	19.23adv 1584	. 2 |- ((B e. C /\ X = Y) -> (E.c(B C_ c /\ A(Fne i^i Ref)c) -> ARefB))
66	44, 65	impbid 574	1 |- ((B e. C /\ X = Y) -> (ARefB <-> E.c(B C_ c /\ A(Fne i^i Ref)c)))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   i^i cin 2592   C_ wss 2593  U.cuni 3177   class class class wbr 3338  Fnecfne 15457  Refcref 15458

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-3an 860  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  df-ref 15464

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