Theorem isushgrm 15952

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 15950	. . 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 2297	. 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 5642	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuhgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2281	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4935	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2234	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4934	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2279	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5642	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuhgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2281	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3658	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2795	. . . 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 5578	. . 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 15898	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2805	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5642	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 15899	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2805	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5642	. . . . . . 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 4935	. . . . . . 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 3658	. . . . . . . . 9 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2795	. . . . . . . 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 5578	. . . . . 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 3068	. . . . 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 3068	. . . 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 2355	. . 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 2952	. 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 1397   E.wex 1540   e. wcel 2201   {cab 2216   {crab 2513   _Vcvv 2801   [.wsbc 3030   ~Pcpw 3653   dom cdm 4727   -1-1->wf1 5325   ` cfv 5328  Vtxcvtx 15892  iEdgciedg 15893  USHGraphcushgr 15948

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-ushgrm 15950

This theorem is referenced by:  ushgrfm  15954  uspgrushgr  16060