Theorem isushgrm 16296

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 16294	. . 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 2305	. 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 5693	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuhgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2289	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4981	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2242	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4980	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2287	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5693	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuhgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2289	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3693	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2815	. . . 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 5629	. . 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 16242	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2825	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5693	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 16243	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2825	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5693	. . . . . . 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 4981	. . . . . . 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 3693	. . . . . . . . 9 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2815	. . . . . . . 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 5629	. . . . . 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 3089	. . . . 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 3089	. . . 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 2365	. . 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 2973	. 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 1402   E.wex 1545   e. wcel 2209   {cab 2224   {crab 2532   _Vcvv 2821   [.wsbc 3051   ~Pcpw 3688   dom cdm 4772   -1-1->wf1 5372   ` cfv 5375  Vtxcvtx 16236  iEdgciedg 16237  USHGraphcushgr 16292

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-ushgrm 16294

This theorem is referenced by:  ushgrfm  16298  uspgrushgr  16404