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Theorem issubgr 28495
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 6881 . . . . . . 7 (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
21adantr 482 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3 fveq2 6881 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantl 483 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
52, 4sseq12d 4013 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6 fveq2 6881 . . . . . . 7 (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
76adantr 482 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8 fveq2 6881 . . . . . . . 8 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
98adantl 483 . . . . . . 7 ((𝑠 = 𝑆𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
106dmeqd 5900 . . . . . . . 8 (𝑠 = 𝑆 → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
1110adantr 482 . . . . . . 7 ((𝑠 = 𝑆𝑔 = 𝐺) → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
129, 11reseq12d 5977 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
137, 12eqeq12d 2749 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
14 fveq2 6881 . . . . . . 7 (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
151pweqd 4615 . . . . . . 7 (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
1614, 15sseq12d 4013 . . . . . 6 (𝑠 = 𝑆 → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
1716adantr 482 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
185, 13, 173anbi123d 1437 . . . 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 28492 . . . 4 SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
2018, 19brabga 5530 . . 3 ((𝑆𝑈𝐺𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
2120ancoms 460 . 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 4010 . . 3 (𝑉𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25 issubgr.i . . . 4 𝐼 = (iEdg‘𝑆)
26 issubgr.b . . . . 5 𝐵 = (iEdg‘𝐺)
2725dmeqi 5899 . . . . 5 dom 𝐼 = dom (iEdg‘𝑆)
2826, 27reseq12i 5974 . . . 4 (𝐵 ↾ dom 𝐼) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))
2925, 28eqeq12i 2751 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
30 issubgr.e . . . 4 𝐸 = (Edg‘𝑆)
3122pweqi 4614 . . . 4 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
3230, 31sseq12i 4010 . . 3 (𝐸 ⊆ 𝒫 𝑉 ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
3324, 29, 323anbi123i 1156 . 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, 33bitr4di 289 1 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wss 3946  𝒫 cpw 4598   class class class wbr 5144  dom cdm 5672  cres 5674  cfv 6535  Vtxcvtx 28223  iEdgciedg 28224  Edgcedg 28274   SubGraph csubgr 28491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-xp 5678  df-dm 5682  df-res 5684  df-iota 6487  df-fv 6543  df-subgr 28492
This theorem is referenced by:  issubgr2  28496  subgrprop  28497  uhgrissubgr  28499  egrsubgr  28501  0grsubgr  28502  uhgrspan1  28527
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