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Theorem issubgr 26505
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 6411 . . . . . . 7 (𝑠 = 𝑆 → (Vtx‘𝑠) = (Vtx‘𝑆))
21adantr 473 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (Vtx‘𝑠) = (Vtx‘𝑆))
3 fveq2 6411 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43adantl 474 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (Vtx‘𝑔) = (Vtx‘𝐺))
52, 4sseq12d 3830 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)))
6 fveq2 6411 . . . . . . 7 (𝑠 = 𝑆 → (iEdg‘𝑠) = (iEdg‘𝑆))
76adantr 473 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → (iEdg‘𝑠) = (iEdg‘𝑆))
8 fveq2 6411 . . . . . . . 8 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
98adantl 474 . . . . . . 7 ((𝑠 = 𝑆𝑔 = 𝐺) → (iEdg‘𝑔) = (iEdg‘𝐺))
106dmeqd 5529 . . . . . . . 8 (𝑠 = 𝑆 → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
1110adantr 473 . . . . . . 7 ((𝑠 = 𝑆𝑔 = 𝐺) → dom (iEdg‘𝑠) = dom (iEdg‘𝑆))
129, 11reseq12d 5601 . . . . . 6 ((𝑠 = 𝑆𝑔 = 𝐺) → ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
137, 12eqeq12d 2814 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
14 fveq2 6411 . . . . . . 7 (𝑠 = 𝑆 → (Edg‘𝑠) = (Edg‘𝑆))
151pweqd 4354 . . . . . . 7 (𝑠 = 𝑆 → 𝒫 (Vtx‘𝑠) = 𝒫 (Vtx‘𝑆))
1614, 15sseq12d 3830 . . . . . 6 (𝑠 = 𝑆 → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
1716adantr 473 . . . . 5 ((𝑠 = 𝑆𝑔 = 𝐺) → ((Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠) ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
185, 13, 173anbi123d 1561 . . . 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 26502 . . . 4 SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
2018, 19brabga 5185 . . 3 ((𝑆𝑈𝐺𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
2120ancoms 451 . 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 3827 . . 3 (𝑉𝐴 ↔ (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
25 issubgr.i . . . 4 𝐼 = (iEdg‘𝑆)
26 issubgr.b . . . . 5 𝐵 = (iEdg‘𝐺)
2725dmeqi 5528 . . . . 5 dom 𝐼 = dom (iEdg‘𝑆)
2826, 27reseq12i 5598 . . . 4 (𝐵 ↾ dom 𝐼) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))
2925, 28eqeq12i 2813 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) ↔ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
30 issubgr.e . . . 4 𝐸 = (Edg‘𝑆)
3122pweqi 4353 . . . 4 𝒫 𝑉 = 𝒫 (Vtx‘𝑆)
3230, 31sseq12i 3827 . . 3 (𝐸 ⊆ 𝒫 𝑉 ↔ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
3324, 29, 323anbi123i 1195 . 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 281 1 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wss 3769  𝒫 cpw 4349   class class class wbr 4843  dom cdm 5312  cres 5314  cfv 6101  Vtxcvtx 26231  iEdgciedg 26232  Edgcedg 26282   SubGraph csubgr 26501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-dm 5322  df-res 5324  df-iota 6064  df-fv 6109  df-subgr 26502
This theorem is referenced by:  issubgr2  26506  subgrprop  26507  uhgrissubgr  26509  egrsubgr  26511  0grsubgr  26512  uhgrspan1  26537
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