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Theorem subgrprop3 40500
 Description: The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺. (Contributed by AV, 19-Nov-2020.)
Hypotheses
Ref Expression
subgrprop3.v 𝑉 = (Vtx‘𝑆)
subgrprop3.a 𝐴 = (Vtx‘𝐺)
subgrprop3.e 𝐸 = (Edg‘𝑆)
subgrprop3.b 𝐵 = (Edg‘𝐺)
Assertion
Ref Expression
subgrprop3 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))

Proof of Theorem subgrprop3
StepHypRef Expression
1 subgrprop3.v . . . 4 𝑉 = (Vtx‘𝑆)
2 subgrprop3.a . . . 4 𝐴 = (Vtx‘𝐺)
3 eqid 2610 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2610 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 subgrprop3.e . . . 4 𝐸 = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 40498 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 3simpa 1051 . . 3 ((𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
86, 7syl 17 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
9 simprl 790 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉𝐴)
10 rnss 5275 . . . . . 6 ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
1110adantl 481 . . . . 5 ((𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
1211adantl 481 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
13 subgrv 40494 . . . . . 6 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
14 edgaval 25794 . . . . . . . . 9 (𝑆 ∈ V → (Edg‘𝑆) = ran (iEdg‘𝑆))
1514adantr 480 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
165, 15syl5eq 2656 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
17 subgrprop3.b . . . . . . . 8 𝐵 = (Edg‘𝐺)
18 edgaval 25794 . . . . . . . . 9 (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
1918adantl 481 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
2017, 19syl5eq 2656 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
2116, 20sseq12d 3597 . . . . . 6 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2213, 21syl 17 . . . . 5 (𝑆 SubGraph 𝐺 → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2322adantr 480 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2412, 23mpbird 246 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸𝐵)
259, 24jca 553 . 2 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉𝐴𝐸𝐵))
268, 25mpdan 699 1 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  ran crn 5039  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674  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-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-edga 25793  df-subgr 40492 This theorem is referenced by: (None)
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