Theorem subumgredg2

Description: An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020)

	Ref	Expression
Hypotheses	subumgredg2.v	⊢ 𝑉 = ( Vtx ‘ 𝑆 )
	subumgredg2.i	⊢ 𝐼 = ( iEdg ‘ 𝑆 )
Assertion	subumgredg2	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ { 𝑒 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑒 ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		subumgredg2.v	⊢ 𝑉 = ( Vtx ‘ 𝑆 )
2		subumgredg2.i	⊢ 𝐼 = ( iEdg ‘ 𝑆 )
3		fveqeq2	⊢ ( 𝑒 = ( 𝐼 ‘ 𝑋 ) → ( ( ♯ ‘ 𝑒 ) = 2 ↔ ( ♯ ‘ ( 𝐼 ‘ 𝑋 ) ) = 2 ) )
4		umgruhgr	⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
5	4	3ad2ant2	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝐺 ∈ UHGraph )
6		simp1	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝑆 SubGraph 𝐺 )
7		simp3	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ dom 𝐼 )
8	1 2 5 6 7	subgruhgredgd	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) )
9		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
10	9	uhgrfun	⊢ ( 𝐺 ∈ UHGraph → Fun ( iEdg ‘ 𝐺 ) )
11	4 10	syl	⊢ ( 𝐺 ∈ UMGraph → Fun ( iEdg ‘ 𝐺 ) )
12	11	3ad2ant2	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → Fun ( iEdg ‘ 𝐺 ) )
13		eqid	⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ 𝑆 )
14		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
15		eqid	⊢ ( Edg ‘ 𝑆 ) = ( Edg ‘ 𝑆 )
16	13 14 2 9 15	subgrprop2	⊢ ( 𝑆 SubGraph 𝐺 → ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) )
17	16	simp2d	⊢ ( 𝑆 SubGraph 𝐺 → 𝐼 ⊆ ( iEdg ‘ 𝐺 ) )
18	17	3ad2ant1	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝐼 ⊆ ( iEdg ‘ 𝐺 ) )
19		funssfv	⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) = ( 𝐼 ‘ 𝑋 ) )
20	19	eqcomd	⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) )
21	12 18 7 20	syl3anc	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) )
22	21	fveq2d	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( ♯ ‘ ( 𝐼 ‘ 𝑋 ) ) = ( ♯ ‘ ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) )
23		simp2	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝐺 ∈ UMGraph )
24	2	dmeqi	⊢ dom 𝐼 = dom ( iEdg ‘ 𝑆 )
25	24	eleq2i	⊢ ( 𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) )
26		subgreldmiedg	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )
27	26	ex	⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) )
28	25 27	syl5bi	⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) )
29	28	a1d	⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐺 ∈ UMGraph → ( 𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) ) )
30	29	3imp	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )
31	14 9	umgredg2	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ♯ ‘ ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) = 2 )
32	23 30 31	syl2anc	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( ♯ ‘ ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) = 2 )
33	22 32	eqtrd	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( ♯ ‘ ( 𝐼 ‘ 𝑋 ) ) = 2 )
34	3 8 33	elrabd	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ { 𝑒 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑒 ) = 2 } )
35		prprrab	⊢ { 𝑒 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑒 ) = 2 } = { 𝑒 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑒 ) = 2 }
36	34 35	eleqtrdi	⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ { 𝑒 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑒 ) = 2 } )