Theorem ishgrag 10740

Description: Express the predicate "H is a hypergraph." Definition of hypergraph in [Diestel] p. 28, which states "A hypergraph is a pair (V, E) of disjoint sets, where the elements of E are non-empty subsets (of any cardinality) of V."

Because V and E are both used as symbols (for the universal class df-v 1815 and the epsilon relation df-eprel 2838, respectively) in Metamath, we instead use A to represent V, the set of vertices or atoms of the hypergraph, and B to represent E, the set of edges or blocks that each connect one or more atoms in A. (See Definition 2.1 in [McKMegPav] p. 2384 for an analogous use of atom and block in Greechie diagrams.)

Hypothesis

Ref	Expression
ishgrag.1	|- H = <.A, B>.

Assertion

Ref	Expression
ishgrag	|- ((A e. C /\ B e. D) -> (H e. HypGrph <-> ((A i^i B) = (/) /\ A.b e. B (b (_ A /\ b =/= (/)))))

Distinct variable groups:   A,b   B,b

Proof of Theorem ishgrag

Step	Hyp	Ref	Expression
1		ineq1 2213	. . . . . . 7 |- (x = A -> (x i^i y) = (A i^i y))
2	1	eqeq1d 1486	. . . . . 6 |- (x = A -> ((x i^i y) = (/) <-> (A i^i y) = (/)))
3		pweq 2407	. . . . . . . 8 |- (x = A -> P~x = P~A)
4	3	difeq1d 2161	. . . . . . 7 |- (x = A -> (P~x \ {(/)}) = (P~A \ {(/)}))
5	4	sseq2d 2092	. . . . . 6 |- (x = A -> (y (_ (P~x \ {(/)}) <-> y (_ (P~A \ {(/)})))
6	2, 5	anbi12d 630	. . . . 5 |- (x = A -> (((x i^i y) = (/) /\ y (_ (P~x \ {(/)})) <-> ((A i^i y) = (/) /\ y (_ (P~A \ {(/)}))))
7		ineq2 2214	. . . . . . 7 |- (y = B -> (A i^i y) = (A i^i B))
8	7	eqeq1d 1486	. . . . . 6 |- (y = B -> ((A i^i y) = (/) <-> (A i^i B) = (/)))
9		sseq1 2085	. . . . . 6 |- (y = B -> (y (_ (P~A \ {(/)}) <-> B (_ (P~A \ {(/)})))
10	8, 9	anbi12d 630	. . . . 5 |- (y = B -> (((A i^i y) = (/) /\ y (_ (P~A \ {(/)})) <-> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)}))))
11	6, 10	opelopabg 2823	. . . 4 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ((x i^i y) = (/) /\ y (_ (P~x \ {(/)}))} <-> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)}))))
12		df-hgra 10737	. . . . 5 |- HypGrph = {<.x, y>. | ((x i^i y) = (/) /\ y (_ (P~x \ {(/)}))}
13	12	eleq2i 1541	. . . 4 |- (<.A, B>. e. HypGrph <-> <.A, B>. e. {<.x, y>. | ((x i^i y) = (/) /\ y (_ (P~x \ {(/)}))})
14	11, 13	syl5bb 534	. . 3 |- ((A e. C /\ B e. D) -> (<.A, B>. e. HypGrph <-> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)}))))
15		ishgrag.1	. . . 4 |- H = <.A, B>.
16	15	eleq1i 1540	. . 3 |- (H e. HypGrph <-> <.A, B>. e. HypGrph)
17	14, 16	syl5bb 534	. 2 |- ((A e. C /\ B e. D) -> (H e. HypGrph <-> ((A i^i B) = (/) /\ B (_ (P~A \ {(/)}))))
18		blkssatm 10738	. . 3 |- (B (_ (P~A \ {(/)}) <-> A.b e. B (b (_ A /\ b =/= (/)))
19	18	anbi2i 482	. 2 |- (((A i^i B) = (/) /\ B (_ (P~A \ {(/)})) <-> ((A i^i B) = (/) /\ A.b e. B (b (_ A /\ b =/= (/))))
20	17, 19	syl6bb 538	1 |- ((A e. C /\ B e. D) -> (H e. HypGrph <-> ((A i^i B) = (/) /\ A.b e. B (b (_ A /\ b =/= (/)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648   \ cdif 2047   i^i cin 2049   (_ wss 2050  (/)c0 2283  P~cpw 2405  {csn 2413  <.cop 2415  {copab 2671  HypGrphchgra 10736

This theorem is referenced by:  emhgrat 10746

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-hgra 10737

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