Theorem subusgr 29875

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 2760	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2760	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2760	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2760	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2760	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29860	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		usgruhgr 29772	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 29868	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
9	7, 8	sylan 592	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
10	9	ancoms 464	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → Fun (iEdg‘𝑆))
11	10	funfnd 6571	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
12	11	adantl 487	. . . . . . 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 789	. . . . . . . . . 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 29767	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
15	14	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ UMGraph)
16	15	adantl 487	. . . . . . . . . . 11 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ UMGraph)
17	16	adantr 486	. . . . . . . . . 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 490	. . . . . . . . . 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 29871	. . . . . . . . . 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 1398	. . . . . . . . 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 3154	. . . . . . . 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 7121	. . . . . . . 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 596	. . . . . . 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 6542	. . . . . . 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 595	. . . . . 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 1155	. . . . . . . . 9 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
27	2, 4	usgrfs 29743	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2})
28		df-f1 6543	. . . . . . . . . . . 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 6712	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} → Fun (iEdg‘𝐺))
30	29	anim1i 627	. . . . . . . . . . . 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 220	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
32	27, 31	syl 18	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
33	32	adantl 487	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺)))
34	26, 33	anim12ci 626	. . . . . . . 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 1105	. . . . . . . 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 237	. . . . . . 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 6857	. . . . . . 7 ⊢ ((Fun (iEdg‘𝐺) ∧ Fun ◡(iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)) → Fun ◡(iEdg‘𝑆))
38	36, 37	syl 18	. . . . . 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 6543	. . . . . 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 595	. . . . 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 29856	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
42	1, 3	isusgrs 29742	. . . . . . . . 9 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
43	42	adantr 486	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
44	41, 43	syl 18	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
45	44	adantr 486	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
46	45	adantl 487	. . . . 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 260	. . . 4 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph)) → 𝑆 ∈ USGraph)
48	47	ex 418	. . 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 18	. 2 ⊢ (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ USGraph) → 𝑆 ∈ USGraph))
50	49	anabsi8 685	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895  𝒫 cpw 4556   class class class wbr 5102  ^◡ccnv 5649  dom cdm 5650  ran crn 5651  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  2c2 12400  ♯chash 14477  Vtxcvtx 29579  iEdgciedg 29580  Edgcedg 29630  UHGraphcuhgr 29639  UMGraphcumgr 29664  USGraphcusgr 29735   SubGraph csubgr 29853

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-n0 12610  df-z 12697  df-uz 12969  df-fz 13643  df-hash 14478  df-edg 29631  df-uhgr 29641  df-upgr 29665  df-umgr 29666  df-uspgr 29736  df-usgr 29737  df-subgr 29854

This theorem is used by:  usgrspan  29881  isubgrusgr  49063