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Theorem issubgr 27047
Description: The property of a set to be a subgraph of another set. (Contributed by AV, 16-Nov-2020.)
Hypotheses
Ref Expression
issubgr.v 𝑉 = (Vtx‘𝑆)
issubgr.a 𝐴 = (Vtx‘𝐺)
issubgr.i 𝐼 = (iEdg‘𝑆)
issubgr.b 𝐵 = (iEdg‘𝐺)
issubgr.e 𝐸 = (Edg‘𝑆)
Assertion
Ref Expression
issubgr ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))

Proof of Theorem issubgr
Dummy variables 𝑠 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6665 . . . . . . 7 (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
21adantr 483 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3 fveq2 6665 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantl 484 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
52, 4sseq12d 4000 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6 fveq2 6665 . . . . . . 7 (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
76adantr 483 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8 fveq2 6665 . . . . . . . 8 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
98adantl 484 . . . . . . 7 ((𝑠 = 𝑆𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
106dmeqd 5769 . . . . . . . 8 (𝑠 = 𝑆 → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
1110adantr 483 . . . . . . 7 ((𝑠 = 𝑆𝑔 = 𝐺) → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
129, 11reseq12d 5849 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
137, 12eqeq12d 2837 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
14 fveq2 6665 . . . . . . 7 (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
151pweqd 4544 . . . . . . 7 (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
1614, 15sseq12d 4000 . . . . . 6 (𝑠 = 𝑆 → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
1716adantr 483 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
185, 13, 173anbi123d 1432 . . . 4 ((𝑠 = 𝑆𝑔 = 𝐺) → (((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠)) ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
19 df-subgr 27044 . . . 4 SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
2018, 19brabga 5414 . . 3 ((𝑆𝑈𝐺𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
2120ancoms 461 . 2 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
22 issubgr.v . . . 4 𝑉 = (Vtx‘𝑆)
23 issubgr.a . . . 4 𝐴 = (Vtx‘𝐺)
2422, 23sseq12i 3997 . . 3 (𝑉𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25 issubgr.i . . . 4 𝐼 = (iEdg‘𝑆)
26 issubgr.b . . . . 5 𝐵 = (iEdg‘𝐺)
2725dmeqi 5768 . . . . 5 dom 𝐼 = dom (iEdg‘𝑆)
2826, 27reseq12i 5846 . . . 4 (𝐵 ↾ dom 𝐼) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))
2925, 28eqeq12i 2836 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
30 issubgr.e . . . 4 𝐸 = (Edg‘𝑆)
3122pweqi 4543 . . . 4 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
3230, 31sseq12i 3997 . . 3 (𝐸 ⊆ 𝒫 𝑉 ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
3324, 29, 323anbi123i 1151 . 2 ((𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
3421, 33syl6bbr 291 1 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wss 3936  𝒫 cpw 4539   class class class wbr 5059  dom cdm 5550  cres 5552  cfv 6350  Vtxcvtx 26775  iEdgciedg 26776  Edgcedg 26826   SubGraph csubgr 27043
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-xp 5556  df-dm 5560  df-res 5562  df-iota 6309  df-fv 6358  df-subgr 27044
This theorem is referenced by:  issubgr2  27048  subgrprop  27049  uhgrissubgr  27051  egrsubgr  27053  0grsubgr  27054  uhgrspan1  27079
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