Theorem isushgrm 15916

Description: The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)

Hypotheses

Ref	Expression
isuhgr.v	|- V = (Vtx ` G )
isuhgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
isushgrm	|- ( G e. U -> ( G e. USHGraph <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )

Distinct variable groups:   j, s   V, s

Allowed substitution hints:   U( j, s)   E( j, s)   G( j, s)   V( j)

Proof of Theorem isushgrm

Dummy variables g h v e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ushgrm 15914	. . 3 |- USHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } }
2	1	eleq2i 2296	. 2 |- ( G e. USHGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } } )
3		fveq2 5635	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuhgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2280	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4931	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2233	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4930	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2278	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5635	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuhgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2280	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3655	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2794	. . . 4 |- ( h = G -> { s e. ~P (Vtx ` h ) | E. j j e. s } = { s e. ~P V | E. j j e. s } )
15	5, 9, 14	f1eq123d 5572	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )
16		vtxex 15862	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2804	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5635	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 15863	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2804	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5635	. . . . . . 7 |- ( g = h -> (iEdg ` g ) = (iEdg ` h ) )
24	23	adantr 276	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) = (iEdg ` h ) )
25		simpr 110	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> e = (iEdg ` h ) )
26	25	dmeqd 4931	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		simpr 110	. . . . . . . . . 10 |- ( ( g = h /\ v = (Vtx ` h ) ) -> v = (Vtx ` h ) )
28	27	pweqd 3655	. . . . . . . . 9 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2794	. . . . . . . 8 |- ( ( g = h /\ v = (Vtx ` h ) ) -> { s e. ~P v | E. j j e. s } = { s e. ~P (Vtx ` h ) | E. j j e. s } )
30	29	adantr 276	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> { s e. ~P v | E. j j e. s } = { s e. ~P (Vtx ` h ) | E. j j e. s } )
31	25, 26, 30	f1eq123d 5572	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e -1-1-> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
32	22, 24, 31	sbcied2 3067	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
33	18, 19, 32	sbcied2 3067	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
34	33	cbvabv 2354	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } } = { h | (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { s e. ~P (Vtx ` h ) | E. j j e. s } }
35	15, 34	elab2g 2951	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } } <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )
36	2, 35	bitrid 192	1 |- ( G e. U -> ( G e. USHGraph <-> E : dom E -1-1-> { s e. ~P V | E. j j e. s } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   {crab 2512   _Vcvv 2800   [.wsbc 3029   ~Pcpw 3650   dom cdm 4723   -1-1->wf1 5321   ` cfv 5324  Vtxcvtx 15856  iEdgciedg 15857  USHGraphcushgr 15912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sub 8345  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-dec 9605  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-ushgrm 15914

This theorem is referenced by:  ushgrfm  15918  uspgrushgr  16024