Theorem ispgrag 16301

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

Because V and E are both used as symbols (for the universal class df-v 2294 and the epsilon relation df-eprel 3583, 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 C_ (P u. (P ^m 2o)))))

Proof of Theorem ispgrag

Step	Hyp	Ref	Expression
1		ineq1 2789	. . . . 5 |- (a = P -> (a i^i b) = (P i^i b))
2	1	eqeq1d 1892	. . . 4 |- (a = P -> ((a i^i b) = (/) <-> (P i^i b) = (/)))
3		id 73	. . . . . 6 |- (a = P -> a = P)
4		opreq1 4889	. . . . . 6 |- (a = P -> (a ^m 2o) = (P ^m 2o))
5	3, 4	uneq12d 2756	. . . . 5 |- (a = P -> (a u. (a ^m 2o)) = (P u. (P ^m 2o)))
6	5	sseq2d 2645	. . . 4 |- (a = P -> (b C_ (a u. (a ^m 2o)) <-> b C_ (P u. (P ^m 2o))))
7	2, 6	anbi12d 690	. . 3 |- (a = P -> (((a i^i b) = (/) /\ b C_ (a u. (a ^m 2o))) <-> ((P i^i b) = (/) /\ b C_ (P u. (P ^m 2o)))))
8		ineq2 2790	. . . . 5 |- (b = L -> (P i^i b) = (P i^i L))
9	8	eqeq1d 1892	. . . 4 |- (b = L -> ((P i^i b) = (/) <-> (P i^i L) = (/)))
10		sseq1 2637	. . . 4 |- (b = L -> (b C_ (P u. (P ^m 2o)) <-> L C_ (P u. (P ^m 2o))))
11	9, 10	anbi12d 690	. . 3 |- (b = L -> (((P i^i b) = (/) /\ b C_ (P u. (P ^m 2o))) <-> ((P i^i L) = (/) /\ L C_ (P u. (P ^m 2o)))))
12	7, 11	opelopabg 3567	. 2 |- ((P e. A /\ L e. B) -> (<.P, L>. e. {<.a, b>. | ((a i^i b) = (/) /\ b C_ (a u. (a ^m 2o)))} <-> ((P i^i L) = (/) /\ L C_ (P u. (P ^m 2o)))))
13		ispgrag.1	. . 3 |- G = <.P, L>.
14		df-pgra 16300	. . 3 |- PsGrph = {<.a, b>. | ((a i^i b) = (/) /\ b C_ (a u. (a ^m 2o)))}
15	13, 14	eleq12i 1962	. 2 |- (G e. PsGrph <-> <.P, L>. e. {<.a, b>. | ((a i^i b) = (/) /\ b C_ (a u. (a ^m 2o)))})
16	12, 15	syl5bb 591	1 |- ((P e. A /\ L e. B) -> (G e. PsGrph <-> ((P i^i L) = (/) /\ L C_ (P u. (P ^m 2o)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  <.cop 3046  {copab 3395  (class class class)co 4884  2oc2o 5173   ^m cmap 5381  PsGrphcpgra 16299

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-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

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-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-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-pgra 16300

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