Theorem subumgredg2en 16266

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
subumgredg2en	⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ 𝑒 ≈ 2o})

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

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

Proof of Theorem subumgredg2en

Dummy variables 𝑗 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4112	. 2 ⊢ (𝑒 = (𝐼‘𝑋) → (𝑒 ≈ 2o ↔ (𝐼‘𝑋) ≈ 2o))
2		subumgredg2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝑆)
3		subumgredg2.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝑆)
4		umgruhgr 16108	. . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
5	4	3ad2ant2 1046	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UHGraph)
6		simp1 1024	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑆 SubGraph 𝐺)
7		simp3 1026	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
8	2, 3, 5, 6, 7	subgruhgredgdm 16265	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
9		elrabi 2970	. . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} → (𝐼‘𝑋) ∈ 𝒫 𝑉)
10	8, 9	syl 14	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ 𝒫 𝑉)
11		eqid 2232	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
12	11	uhgrfun 16072	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
13	4, 12	syl 14	. . . . 5 ⊢ (𝐺 ∈ UMGraph → Fun (iEdg‘𝐺))
14	13	3ad2ant2 1046	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → Fun (iEdg‘𝐺))
15		eqid 2232	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
16		eqid 2232	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
17		eqid 2232	. . . . . . 7 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
18	15, 16, 3, 11, 17	subgrprop2 16255	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
19	18	simp2d 1037	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → 𝐼 ⊆ (iEdg‘𝐺))
20	19	3ad2ant1 1045	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 ⊆ (iEdg‘𝐺))
21		funssfv 5696	. . . . 5 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
22	21	eqcomd 2238	. . . 4 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
23	14, 20, 7, 22	syl3anc 1274	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋))
24		simp2 1025	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UMGraph)
25	3	dmeqi 4957	. . . . . . . 8 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
26	25	eleq2i 2299	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom (iEdg‘𝑆))
27		subgreldmiedg 16264	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))
28	27	ex 115	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom (iEdg‘𝑆) → 𝑋 ∈ dom (iEdg‘𝐺)))
29	26, 28	biimtrid 152	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom (iEdg‘𝐺)))
30	29	a1d 22	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → (𝐺 ∈ UMGraph → (𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom (iEdg‘𝐺))))
31	30	3imp 1220	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom (iEdg‘𝐺))
32	16, 11	umgredg2en 16104	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑋) ≈ 2o)
33	24, 31, 32	syl2anc 411	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) ≈ 2o)
34	23, 33	eqbrtrd 4131	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ≈ 2o)
35	1, 10, 34	elrabd 2975	1 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ 𝑒 ≈ 2o})

Syntax hints:   → wi 4   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  {crab 2524   ⊆ wss 3211  𝒫 cpw 3669   class class class wbr 4109  dom cdm 4749  Fun wfun 5346  ‘cfv 5352  2_oc2o 6641   ≈ cen 6973  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052  UHGraphcuhgr 16062  UMGraphcumgr 16087   SubGraph csubgr 16248

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-1o 6647  df-2o 6648  df-en 6976  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-edg 16053  df-uhgrm 16064  df-upgren 16088  df-umgren 16089  df-subgr 16249

This theorem is referenced by:  subumgr  16269  subusgr  16270