Theorem uhgrvtxedgiedgb 26031

Description: In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.)

Hypotheses

Ref	Expression
uhgrvtxedgiedgb.v	|- V = (Vtx ` G )
uhgrvtxedgiedgb.i	|- I = (iEdg ` G )
uhgrvtxedgiedgb.e	|- E = (Edg ` G )

Assertion

Ref	Expression
uhgrvtxedgiedgb	|- ( ( G e. UHGraph /\ U e. V ) -> ( E. i e. dom I U e. ( I ` i ) <-> E. e e. E U e. e ) )

Distinct variable groups:   e, E   e, I, i   U, e, i

Allowed substitution hints:   E( i)   G( e, i)   V( e, i)

Proof of Theorem uhgrvtxedgiedgb

Step	Hyp	Ref	Expression
1		edgval 25941	. . . . . . 7 |- (Edg ` G ) = ran (iEdg ` G )
2	1	a1i 11	. . . . . 6 |- ( G e. UHGraph -> (Edg ` G ) = ran (iEdg ` G ) )
3		uhgrvtxedgiedgb.e	. . . . . 6 |- E = (Edg ` G )
4		uhgrvtxedgiedgb.i	. . . . . . 7 |- I = (iEdg ` G )
5	4	rneqi 5352	. . . . . 6 |- ran I = ran (iEdg ` G )
6	2, 3, 5	3eqtr4g 2681	. . . . 5 |- ( G e. UHGraph -> E = ran I )
7	6	rexeqdv 3145	. . . 4 |- ( G e. UHGraph -> ( E. e e. E U e. e <-> E. e e. ran I U e. e ) )
8	4	uhgrfun 25961	. . . . . 6 |- ( G e. UHGraph -> Fun I )
9		funfn 5918	. . . . . 6 |- ( Fun I <-> I Fn dom I )
10	8, 9	sylib 208	. . . . 5 |- ( G e. UHGraph -> I Fn dom I )
11		eleq2 2690	. . . . . 6 |- ( e = ( I ` i ) -> ( U e. e <-> U e. ( I ` i ) ) )
12	11	rexrn 6361	. . . . 5 |- ( I Fn dom I -> ( E. e e. ran I U e. e <-> E. i e. dom I U e. ( I ` i ) ) )
13	10, 12	syl 17	. . . 4 |- ( G e. UHGraph -> ( E. e e. ran I U e. e <-> E. i e. dom I U e. ( I ` i ) ) )
14	7, 13	bitrd 268	. . 3 |- ( G e. UHGraph -> ( E. e e. E U e. e <-> E. i e. dom I U e. ( I ` i ) ) )
15	14	adantr 481	. 2 |- ( ( G e. UHGraph /\ U e. V ) -> ( E. e e. E U e. e <-> E. i e. dom I U e. ( I ` i ) ) )
16	15	bicomd 213	1 |- ( ( G e. UHGraph /\ U e. V ) -> ( E. i e. dom I U e. ( I ` i ) <-> E. e e. E U e. e ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-edg 25940  df-uhgr 25953

This theorem is referenced by:  vtxduhgr0edgnel  26390