Theorem subusgr 16516

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 2238	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2238	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2238	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2238	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2238	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 16501	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		usgruhgr 16430	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 16509	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
9	7, 8	sylan 283	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
10	9	ancoms 268	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → Fun (iEdg‘𝑆))
11	10	funfnd 5408	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
12	11	adantl 277	. . . . . . 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 541	. . . . . . . . . 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 16425	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
15	14	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ UMGraph)
16	15	adantl 277	. . . . . . . . . . 11 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ UMGraph)
17	16	adantr 276	. . . . . . . . . 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 110	. . . . . . . . . 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	subumgredg2en 16512	. . . . . . . . . 10 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o})
20	13, 17, 18, 19	syl3anc 1278	. . . . . . . . 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 ≈ 2o})
21	20	ralrimiva 2623	. . . . . . . 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 ≈ 2o})
22		fnfvrnss 5868	. . . . . . . 8 ⊢ (((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o}) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o})
23	12, 21, 22	syl2anc 415	. . . . . . 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 ≈ 2o})
24		df-f 5381	. . . . . . 7 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o} ↔ ((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ran (iEdg‘𝑆) ⊆ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o}))
25	12, 23, 24	sylanbrc 421	. . . . . 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 ≈ 2o})
26		simp2 1029	. . . . . . . . 9 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
27	2, 4	usgrfen 16401	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑦 ≈ 2o})
28		df-f1 5382	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑦 ≈ 2o} ↔ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑦 ≈ 2o} ∧ Fun ◡(iEdg‘𝐺)))
29		ffun 5536	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑦 ≈ 2o} → Fun (iEdg‘𝐺))
30	29	anim1i 340	. . . . . . . . . . . 12 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑦 ≈ 2o} ∧ Fun ◡(iEdg‘𝐺)) → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
31	28, 30	sylbi 121	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑦 ≈ 2o} → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
32	27, 31	syl 14	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
33	32	adantl 277	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
34	26, 33	anim12ci 339	. . . . . . . 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 1011	. . . . . . . 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 134	. . . . . . 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 5671	. . . . . . 7 ⊢ ((Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)) → Fun ◡(iEdg‘𝑆))
38	36, 37	syl 14	. . . . . 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 5382	. . . . . 6 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o} ↔ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o} ∧ Fun ◡(iEdg‘𝑆)))
40	25, 38, 39	sylanbrc 421	. . . . 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 ≈ 2o})
41		subgrv 16497	. . . . . . . . 9 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
42	41	simpld 112	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → 𝑆 ∈ V)
43	1, 3	isusgren 16399	. . . . . . . 8 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o}))
44	42, 43	syl 14	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o}))
45	44	adantr 276	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ 𝑒 ≈ 2o}))
46	45	adantl 277	. . . . 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 ≈ 2o}))
47	40, 46	mpbird 167	. . . 4 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → 𝑆 ∈ USGraph)
48	47	ex 115	. . 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 14	. 2 ⊢ (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → 𝑆 ∈ USGraph))
50	49	anabsi8 588	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688   class class class wbr 4130  ^◡ccnv 4773  dom cdm 4774  ran crn 4775  Fun wfun 5371   Fn wfn 5372  ⟶wf 5373  –1-1→wf1 5374  ‘cfv 5377  2_oc2o 6681   ≈ cen 7020  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  UHGraphcuhgr 16308  UMGraphcumgr 16333  USGraphcusgr 16395   SubGraph csubgr 16494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-upgren 16334  df-umgren 16335  df-uspgren 16396  df-usgren 16397  df-subgr 16495

This theorem is used by:  usgrspan  16522