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Theorem subumgr 40512
 Description: A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.)
Assertion
Ref Expression
subumgr ((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph )

Proof of Theorem subumgr
Dummy variables 𝑥 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
2 eqid 2610 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2610 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2610 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 eqid 2610 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 40498 . . 3 (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7 umgruhgr 25770 . . . . . . . . . 10 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
8 subgruhgrfun 40506 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
97, 8sylan 487 . . . . . . . . 9 ((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
109ancoms 468 . . . . . . . 8 ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ) → Fun (iEdg‘𝑆))
11 funfn 5833 . . . . . . . 8 (Fun (iEdg‘𝑆) ↔ (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
1210, 11sylib 207 . . . . . . 7 ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
1312adantl 481 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
14 simplrl 796 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑆 SubGraph 𝐺)
15 simplrr 797 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UMGraph )
16 simpr 476 . . . . . . . . 9 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝑆))
171, 3subumgredg2 40509 . . . . . . . . 9 ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2})
1814, 15, 16, 17syl3anc 1318 . . . . . . . 8 (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2})
1918ralrimiva 2949 . . . . . . 7 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) → ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2})
20 fnfvrnss 6297 . . . . . . 7 (((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})
2113, 19, 20syl2anc 691 . . . . . 6 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2})
22 df-f 5808 . . . . . 6 ((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}))
2313, 21, 22sylanbrc 695 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2})
24 subgrv 40494 . . . . . . 7 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
251, 3isumgrs 25762 . . . . . . . 8 (𝑆 ∈ V → (𝑆 ∈ UMGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2}))
2625adantr 480 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UMGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2}))
2724, 26syl 17 . . . . . 6 (𝑆 SubGraph 𝐺 → (𝑆 ∈ UMGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2}))
2827ad2antrl 760 . . . . 5 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) → (𝑆 ∈ UMGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (#‘𝑒) = 2}))
2923, 28mpbird 246 . . . 4 ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺𝐺 ∈ UMGraph )) → 𝑆 ∈ UMGraph )
3029ex 449 . . 3 (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ) → 𝑆 ∈ UMGraph ))
316, 30syl 17 . 2 (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺𝐺 ∈ UMGraph ) → 𝑆 ∈ UMGraph ))
3231anabsi8 857 1 ((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   UMGraph cumgr 25748  Edgcedga 25792   SubGraph csubgr 40491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-subgr 40492 This theorem is referenced by:  umgrspan  40518
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