Theorem ispgrag 10750

Description: Express the predicate "G is a pseudograph."

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 P to represent V, the set of vertices or points of the hypergraph, and L to represent E, the set of edges or lines that each connect one or two vertices in P.

Hypothesis

Ref	Expression
ispgrag.1	|- G = <.P, L>.

Assertion

Ref	Expression
ispgrag	|- ((P e. A /\ L e. B) -> (G e. PsGrph <-> ((P i^i L) = (/) /\ L (_ (P u. (P ^m 2o)))))

Proof of Theorem ispgrag

Step	Hyp	Ref	Expression
1		ineq1 2213	. . . . 5 |- (a = P -> (a i^i b) = (P i^i b))
2	1	eqeq1d 1486	. . . 4 |- (a = P -> ((a i^i b) = (/) <-> (P i^i b) = (/)))
3		id 59	. . . . . 6 |- (a = P -> a = P)
4		opreq1 3974	. . . . . 6 |- (a = P -> (a ^m 2o) = (P ^m 2o))
5	3, 4	uneq12d 2188	. . . . 5 |- (a = P -> (a u. (a ^m 2o)) = (P u. (P ^m 2o)))
6	5	sseq2d 2092	. . . 4 |- (a = P -> (b (_ (a u. (a ^m 2o)) <-> b (_ (P u. (P ^m 2o))))
7	2, 6	anbi12d 630	. . 3 |- (a = P -> (((a i^i b) = (/) /\ b (_ (a u. (a ^m 2o))) <-> ((P i^i b) = (/) /\ b (_ (P u. (P ^m 2o)))))
8		ineq2 2214	. . . . 5 |- (b = L -> (P i^i b) = (P i^i L))
9	8	eqeq1d 1486	. . . 4 |- (b = L -> ((P i^i b) = (/) <-> (P i^i L) = (/)))
10		sseq1 2085	. . . 4 |- (b = L -> (b (_ (P u. (P ^m 2o)) <-> L (_ (P u. (P ^m 2o))))
11	9, 10	anbi12d 630	. . 3 |- (b = L -> (((P i^i b) = (/) /\ b (_ (P u. (P ^m 2o))) <-> ((P i^i L) = (/) /\ L (_ (P u. (P ^m 2o)))))
12	7, 11	opelopabg 2823	. 2 |- ((P e. A /\ L e. B) -> (<.P, L>. e. {<.a, b>. | ((a i^i b) = (/) /\ b (_ (a u. (a ^m 2o)))} <-> ((P i^i L) = (/) /\ L (_ (P u. (P ^m 2o)))))
13		ispgrag.1	. . 3 |- G = <.P, L>.
14		df-pgra 10749	. . 3 |- PsGrph = {<.a, b>. | ((a i^i b) = (/) /\ b (_ (a u. (a ^m 2o)))}
15	13, 14	eleq12i 1542	. 2 |- (G e. PsGrph <-> <.P, L>. e. {<.a, b>. | ((a i^i b) = (/) /\ b (_ (a u. (a ^m 2o)))})
16	12, 15	syl5bb 534	1 |- ((P e. A /\ L e. B) -> (G e. PsGrph <-> ((P i^i L) = (/) /\ L (_ (P u. (P ^m 2o)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   u. cun 2048   i^i cin 2049   (_ wss 2050  (/)c0 2283  <.cop 2415  {copab 2671  (class class class)co 3969  2oc2o 4135   ^m cm 4328  PsGrphcpgra 10748

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-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-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-pgra 10749

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