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Theorem subgrprop3 26213
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 2651 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2651 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 subgrprop3.e . . . 4 𝐸 = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 26211 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 3simpa 1078 . . 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 809 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉𝐴)
10 rnss 5386 . . . . 5 ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
1110ad2antll 765 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12 subgrv 26207 . . . . . 6 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13 edgval 25986 . . . . . . . . 9 (Edg‘𝑆) = ran (iEdg‘𝑆)
1413a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
155, 14syl5eq 2697 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16 subgrprop3.b . . . . . . . 8 𝐵 = (Edg‘𝐺)
17 edgval 25986 . . . . . . . . 9 (Edg‘𝐺) = ran (iEdg‘𝐺)
1817a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
1916, 18syl5eq 2697 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
2015, 19sseq12d 3667 . . . . . 6 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2112, 20syl 17 . . . . 5 (𝑆 SubGraph 𝐺 → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2221adantr 480 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2311, 22mpbird 247 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸𝐵)
249, 23jca 553 . 2 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉𝐴𝐸𝐵))
258, 24mpdan 703 1 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191   class class class wbr 4685  ran crn 5144  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984   SubGraph csubgr 26204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-edg 25985  df-subgr 26205
This theorem is referenced by: (None)
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