Theorem ispgrag 10779

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

Because V and E are both used as symbols (for the universal class df-v 1812 and the epsilon relation df-eprel 2832, 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 2210	. . . . 5 |- (a = P -> (a i^i b) = (P i^i b))
2	1	eqeq1d 1483	. . . 4 |- (a = P -> ((a i^i b) = (/) <-> (P i^i b) = (/)))
3		id 59	. . . . . 6 |- (a = P -> a = P)
4		opreq1 3968	. . . . . 6 |- (a = P -> (a ^m 2o) = (P ^m 2o))
5	3, 4	uneq12d 2185	. . . . 5 |- (a = P -> (a u. (a ^m 2o)) = (P u. (P ^m 2o)))
6	5	sseq2d 2089	. . . 4 |- (a = P -> (b (_ (a u. (a ^m 2o)) <-> b (_ (P u. (P ^m 2o))))
7	2, 6	anbi12d 628	. . 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 2211	. . . . 5 |- (b = L -> (P i^i b) = (P i^i L))
9	8	eqeq1d 1483	. . . 4 |- (b = L -> ((P i^i b) = (/) <-> (P i^i L) = (/)))
10		sseq1 2082	. . . 4 |- (b = L -> (b (_ (P u. (P ^m 2o)) <-> L (_ (P u. (P ^m 2o))))
11	9, 10	anbi12d 628	. . 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 2817	. 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 10778	. . 3 |- PsGrph = {<.a, b>. | ((a i^i b) = (/) /\ b (_ (a u. (a ^m 2o)))}
15	13, 14	eleq12i 1539	. 2 |- (G e. PsGrph <-> <.P, L>. e. {<.a, b>. | ((a i^i b) = (/) /\ b (_ (a u. (a ^m 2o)))})
16	12, 15	syl5bb 532	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 956   e. wcel 958   u. cun 2045   i^i cin 2046   (_ wss 2047  (/)c0 2280  <.cop 2411  {copab 2666  (class class class)co 3963  2oc2o 4129   ^m cm 4322  PsGrphcpgra 10777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-pgra 10778

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