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Theorem subgrprop3 29041
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 2726 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2726 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 subgrprop3.e . . . 4 𝐸 = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 29039 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 3simpa 1145 . . 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 768 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉𝐴)
10 rnss 5932 . . . . 5 ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
1110ad2antll 726 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12 subgrv 29035 . . . . . 6 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13 edgval 28817 . . . . . . . . 9 (Edg‘𝑆) = ran (iEdg‘𝑆)
1413a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
155, 14eqtrid 2778 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16 subgrprop3.b . . . . . . . 8 𝐵 = (Edg‘𝐺)
17 edgval 28817 . . . . . . . . 9 (Edg‘𝐺) = ran (iEdg‘𝐺)
1817a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
1916, 18eqtrid 2778 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
2015, 19sseq12d 4010 . . . . . 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 257 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸𝐵)
249, 23jca 511 . 2 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉𝐴𝐸𝐵))
258, 24mpdan 684 1 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3468  wss 3943  𝒫 cpw 4597   class class class wbr 5141  ran crn 5670  cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815   SubGraph csubgr 29032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-fv 6545  df-edg 28816  df-subgr 29033
This theorem is referenced by: (None)
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