Theorem isuhgrm 16197

Description: The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
isuhgrm	|- ( G e. U -> ( G e. UHGraph <-> E : dom E --> { 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 isuhgrm

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

Step	Hyp	Ref	Expression
1		df-uhgrm 16195	. . 3 |- UHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } }
2	1	eleq2i 2301	. 2 |- ( G e. UHGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } } )
3		fveq2 5676	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuhgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2285	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4964	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2238	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4963	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2283	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5676	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuhgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2285	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3680	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2809	. . . 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	feq123d 5505	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) --> { s e. ~P (Vtx ` h ) | E. j j e. s } <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
16		vtxex 16144	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2819	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5676	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 16145	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2819	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5676	. . . . . . 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 4964	. . . . . . 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 3680	. . . . . . . . 9 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2809	. . . . . . . 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	feq123d 5505	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e --> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
32	22, 24, 31	sbcied2 3083	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
33	18, 19, 32	sbcied2 3083	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { s e. ~P (Vtx ` h ) | E. j j e. s } ) )
34	33	cbvabv 2361	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } } = { h | (iEdg ` h ) : dom (iEdg ` h ) --> { s e. ~P (Vtx ` h ) | E. j j e. s } }
35	15, 34	elab2g 2967	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } } <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
36	2, 35	bitrid 192	1 |- ( G e. U -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   {crab 2526   _Vcvv 2815   [.wsbc 3045   ~Pcpw 3675   dom cdm 4755   -->wf 5354   ` cfv 5358  Vtxcvtx 16138  iEdgciedg 16139  UHGraphcuhgr 16193

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-cnre 8255

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-sub 8464  df-inn 9259  df-2 9317  df-3 9318  df-4 9319  df-5 9320  df-6 9321  df-7 9322  df-8 9323  df-9 9324  df-n0 9518  df-dec 9732  df-ndx 13304  df-slot 13305  df-base 13307  df-edgf 16131  df-vtx 16140  df-iedg 16141  df-uhgrm 16195

This theorem is referenced by:  uhgrfm  16199  uhgreq12g  16202  ushgruhgr  16206  isuhgropm  16207  uhgr0e  16208  uhgr0  16211  uhgrun  16212  incistruhgr  16216  upgruhgr  16237  subuhgr  16398