Theorem subumgredg2 26177

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

Hypotheses

Ref	Expression
subumgredg2.v	⊢ 𝑉 = (Vtx‘𝑆)
subumgredg2.i	⊢ 𝐼 = (iEdg‘𝑆)

Assertion

Ref	Expression
subumgredg2	⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})

Distinct variable groups:   𝑒,𝐼   𝑒,𝑉   𝑒,𝑋

Allowed substitution hints:   𝑆(𝑒)   𝐺(𝑒)

Proof of Theorem subumgredg2

Step	Hyp	Ref	Expression
1		subumgredg2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝑆)
2		subumgredg2.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
3		umgruhgr 25999	. . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
4	3	3ad2ant2 1083	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UHGraph )
5		simp1 1061	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑆 SubGraph 𝐺)
6		simp3 1063	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
7	1, 2, 4, 5, 6	subgruhgredgd 26176	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}))
8		eqid 2622	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
9	8	uhgrfun 25961	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
10	3, 9	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ UMGraph → Fun (iEdg‘𝐺))
11	10	3ad2ant2 1083	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → Fun (iEdg‘𝐺))
12		eqid 2622	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
13		eqid 2622	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14		eqid 2622	. . . . . . . . 9 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
15	12, 13, 2, 8, 14	subgrprop2 26166	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
16	15	simp2d 1074	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → 𝐼 ⊆ (iEdg‘𝐺))
17	16	3ad2ant1 1082	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 ⊆ (iEdg‘𝐺))
18		funssfv 6209	. . . . . . 7 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
19	18	eqcomd 2628	. . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
20	11, 17, 6, 19	syl3anc 1326	. . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
21	20	fveq2d 6195	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼‘𝑋)) = (#‘((iEdg‘𝐺)‘𝑋)))
22		simp2 1062	. . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UMGraph )
23	2	dmeqi 5325	. . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
24	23	eleq2i 2693	. . . . . . . 8 ⊢ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom (iEdg‘𝑆))
25		subgreldmiedg 26175	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
26	25	ex 450	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom (iEdg‘𝑆) → 𝑋 ∈ dom (iEdg‘𝐺)))
27	24, 26	syl5bi 232	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom (iEdg‘𝐺)))
28	27	a1d 25	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝐺 ∈ UMGraph → (𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom (iEdg‘𝐺))))
29	28	3imp 1256	. . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom (iEdg‘𝐺))
30	13, 8	umgredg2 25995	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → (#‘((iEdg‘𝐺)‘𝑋)) = 2)
31	22, 29, 30	syl2anc 693	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘((iEdg‘𝐺)‘𝑋)) = 2)
32	21, 31	eqtrd 2656	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼‘𝑋)) = 2)
33		fveq2 6191	. . . . 5 ⊢ (𝑒 = (𝐼‘𝑋) → (#‘𝑒) = (#‘(𝐼‘𝑋)))
34	33	eqeq1d 2624	. . . 4 ⊢ (𝑒 = (𝐼‘𝑋) → ((#‘𝑒) = 2 ↔ (#‘(𝐼‘𝑋)) = 2))
35	34	elrab 3363	. . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑒 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑒) = 2} ↔ ((𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘(𝐼‘𝑋)) = 2))
36	7, 32, 35	sylanbrc 698	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑒) = 2})
37		prprrab 13255	. 2 ⊢ {𝑒 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑒) = 2} = {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}
38	36, 37	syl6eleq 2711	1 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653  dom cdm 5114  Fun wfun 5882  ‘cfv 5888  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   UMGraph cumgr 25976   SubGraph csubgr 26159

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-subgr 26160

This theorem is referenced by:  subumgr  26180  subusgr  26181