Theorem subusgr 29306

Description: A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.)

Assertion

Ref	Expression
subusgr	⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph)

Proof of Theorem subusgr

Dummy variables 𝑥 𝑒 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2737	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2737	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2737	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2737	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29291	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		usgruhgr 29203	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 29299	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
9	7, 8	sylan 580	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
10	9	ancoms 458	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → Fun (iEdg‘𝑆))
11	10	funfnd 6597	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
12	11	adantl 481	. . . . . . 7 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
13		simplrl 777	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑆 SubGraph 𝐺)
14		usgrumgr 29198	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ UMGraph)
16	15	adantl 481	. . . . . . . . . . 11 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ UMGraph)
17	16	adantr 480	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UMGraph)
18		simpr 484	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝑆))
19	1, 3	subumgredg2 29302	. . . . . . . . . 10 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
20	13, 17, 18, 19	syl3anc 1373	. . . . . . . . 9 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
21	20	ralrimiva 3146	. . . . . . . 8 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
22		fnfvrnss 7141	. . . . . . . 8 ⊢ (((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
23	12, 21, 22	syl2anc 584	. . . . . . 7 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
24		df-f 6565	. . . . . . 7 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2} ↔ ((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ran (iEdg‘𝑆) ⊆ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
25	12, 23, 24	sylanbrc 583	. . . . . 6 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
26		simp2 1138	. . . . . . . . 9 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
27	2, 4	usgrfs 29174	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2})
28		df-f1 6566	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} ↔ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} ∧ Fun ◡(iEdg‘𝐺)))
29		ffun 6739	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} → Fun (iEdg‘𝐺))
30	29	anim1i 615	. . . . . . . . . . . 12 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} ∧ Fun ◡(iEdg‘𝐺)) → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
31	28, 30	sylbi 217	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
32	27, 31	syl 17	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
33	32	adantl 481	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
34	26, 33	anim12ci 614	. . . . . . . 8 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → ((Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
35		df-3an 1089	. . . . . . . 8 ⊢ ((Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)) ↔ ((Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
36	34, 35	sylibr 234	. . . . . . 7 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
37		f1ssf1 6880	. . . . . . 7 ⊢ ((Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)) → Fun ◡(iEdg‘𝑆))
38	36, 37	syl 17	. . . . . 6 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → Fun ◡(iEdg‘𝑆))
39		df-f1 6566	. . . . . 6 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2} ↔ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2} ∧ Fun ◡(iEdg‘𝑆)))
40	25, 38, 39	sylanbrc 583	. . . . 5 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
41		subgrv 29287	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
42	1, 3	isusgrs 29173	. . . . . . . . 9 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
43	42	adantr 480	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
44	41, 43	syl 17	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
45	44	adantr 480	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
46	45	adantl 481	. . . . 5 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
47	40, 46	mpbird 257	. . . 4 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → 𝑆 ∈ USGraph)
48	47	ex 412	. . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → 𝑆 ∈ USGraph))
49	6, 48	syl 17	. 2 ⊢ (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → 𝑆 ∈ USGraph))
50	49	anabsi8 672	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600   class class class wbr 5143  ^◡ccnv 5684  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561  2c2 12321  ♯chash 14369  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073  UMGraphcumgr 29098  USGraphcusgr 29166   SubGraph csubgr 29284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-subgr 29285

This theorem is referenced by:  usgrspan  29312  isubgrusgr  47858