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Theorem subusgr 29321
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.
StepHypRef Expression
1 eqid 2735 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
2 eqid 2735 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2735 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2735 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 eqid 2735 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 29306 . . 3 (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7 usgruhgr 29218 . . . . . . . . . . 11 (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
8 subgruhgrfun 29314 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
97, 8sylan 580 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
109ancoms 458 . . . . . . . . 9 ((𝑆 SubGraph 𝐺𝐺 ∈ USGraph) → Fun (iEdg‘𝑆))
1110funfnd 6599 . . . . . . . 8 ((𝑆 SubGraph 𝐺𝐺 ∈ USGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
1211adantl 481 . . . . . . 7 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ 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 𝐺𝐺 ∈ USGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑆 SubGraph 𝐺)
14 usgrumgr 29213 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
1514adantl 481 . . . . . . . . . . . 12 ((𝑆 SubGraph 𝐺𝐺 ∈ USGraph) → 𝐺 ∈ UMGraph)
1615adantl 481 . . . . . . . . . . 11 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → 𝐺 ∈ UMGraph)
1716adantr 480 . . . . . . . . . 10 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ 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 𝐺𝐺 ∈ USGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝑆))
191, 3subumgredg2 29317 . . . . . . . . . 10 ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
2013, 17, 18, 19syl3anc 1370 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
2120ralrimiva 3144 . . . . . . . 8 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ 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})
2312, 21, 22syl2anc 584 . . . . . . 7 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
24 df-f 6567 . . . . . . 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}))
2512, 23, 24sylanbrc 583 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
26 simp2 1136 . . . . . . . . 9 (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
272, 4usgrfs 29189 . . . . . . . . . . 11 (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2})
28 df-f1 6568 . . . . . . . . . . . 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 6740 . . . . . . . . . . . . 13 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} → Fun (iEdg‘𝐺))
3029anim1i 615 . . . . . . . . . . . 12 (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} ∧ Fun (iEdg‘𝐺)) → (Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺)))
3128, 30sylbi 217 . . . . . . . . . . 11 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑦 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑦) = 2} → (Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺)))
3227, 31syl 17 . . . . . . . . . 10 (𝐺 ∈ USGraph → (Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺)))
3332adantl 481 . . . . . . . . 9 ((𝑆 SubGraph 𝐺𝐺 ∈ USGraph) → (Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺)))
3426, 33anim12ci 614 . . . . . . . 8 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → ((Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺)) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
35 df-3an 1088 . . . . . . . 8 ((Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)) ↔ ((Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺)) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
3634, 35sylibr 234 . . . . . . 7 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → (Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
37 f1ssf1 6881 . . . . . . 7 ((Fun (iEdg‘𝐺) ∧ Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)) → Fun (iEdg‘𝑆))
3836, 37syl 17 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → Fun (iEdg‘𝑆))
39 df-f1 6568 . . . . . 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)))
4025, 38, 39sylanbrc 583 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2})
41 subgrv 29302 . . . . . . . 8 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
421, 3isusgrs 29188 . . . . . . . . 9 (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
4342adantr 480 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
4441, 43syl 17 . . . . . . 7 (𝑆 SubGraph 𝐺 → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
4544adantr 480 . . . . . 6 ((𝑆 SubGraph 𝐺𝐺 ∈ USGraph) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
4645adantl 481 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → (𝑆 ∈ USGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (♯‘𝑒) = 2}))
4740, 46mpbird 257 . . . 4 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ USGraph)) → 𝑆 ∈ USGraph)
4847ex 412 . . 3 (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺𝐺 ∈ USGraph) → 𝑆 ∈ USGraph))
496, 48syl 17 . 2 (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺𝐺 ∈ USGraph) → 𝑆 ∈ USGraph))
5049anabsi8 672 1 ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  ccnv 5688  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  wf 6559  1-1wf1 6560  cfv 6563  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  UHGraphcuhgr 29088  UMGraphcumgr 29113  USGraphcusgr 29181   SubGraph csubgr 29299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-edg 29080  df-uhgr 29090  df-upgr 29114  df-umgr 29115  df-uspgr 29182  df-usgr 29183  df-subgr 29300
This theorem is referenced by:  usgrspan  29327  isubgrusgr  47796
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