Theorem subumgredg2en 16195

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 4096	. 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 16037	. . . . 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 16194	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
9		elrabi 2960	. . 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 2231	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
12	11	uhgrfun 16001	. . . . . 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 2231	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
16		eqid 2231	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
17		eqid 2231	. . . . . . 7 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
18	15, 16, 3, 11, 17	subgrprop2 16184	. . . . . 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 5674	. . . . 5 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋))
22	21	eqcomd 2237	. . . 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 4938	. . . . . . . 8 ⊢ dom 𝐼 = dom (iEdg‘𝑆)
26	25	eleq2i 2298	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom (iEdg‘𝑆))
27		subgreldmiedg 16193	. . . . . . . 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 16033	. . . 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 4115	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ≈ 2o)
35	1, 10, 34	elrabd 2965	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 2202  {crab 2515   ⊆ wss 3201  𝒫 cpw 3656   class class class wbr 4093  dom cdm 4731  Fun wfun 5327  ‘cfv 5333  2_oc2o 6619   ≈ cen 6950  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981  UHGraphcuhgr 15991  UMGraphcumgr 16016   SubGraph csubgr 16177

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-en 6953  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-uhgrm 15993  df-upgren 16017  df-umgren 16018  df-subgr 16178

This theorem is referenced by:  subumgr  16198  subusgr  16199