Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isubgr3stgrlem5 Structured version   Visualization version   GIF version

Theorem isubgr3stgrlem5 47937
Description: Lemma 5 for isubgr3stgr 47942. (Contributed by AV, 24-Sep-2025.)
Hypotheses
Ref Expression
isubgr3stgr.v 𝑉 = (Vtx‘𝐺)
isubgr3stgr.u 𝑈 = (𝐺 NeighbVtx 𝑋)
isubgr3stgr.c 𝐶 = (𝐺 ClNeighbVtx 𝑋)
isubgr3stgr.n 𝑁 ∈ ℕ0
isubgr3stgr.s 𝑆 = (StarGr‘𝑁)
isubgr3stgr.w 𝑊 = (Vtx‘𝑆)
isubgr3stgr.e 𝐸 = (Edg‘𝐺)
isubgr3stgr.i 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
isubgr3stgr.h 𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))
Assertion
Ref Expression
isubgr3stgrlem5 ((𝐹:𝐶𝑊𝑌𝐼) → (𝐻𝑌) = (𝐹𝑌))
Distinct variable groups:   𝐶,𝑖   𝑖,𝐹   𝑖,𝐼   𝑖,𝑊   𝑖,𝑌
Allowed substitution hints:   𝑆(𝑖)   𝑈(𝑖)   𝐸(𝑖)   𝐺(𝑖)   𝐻(𝑖)   𝑁(𝑖)   𝑉(𝑖)   𝑋(𝑖)

Proof of Theorem isubgr3stgrlem5
StepHypRef Expression
1 isubgr3stgr.h . . 3 𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))
21a1i 11 . 2 ((𝐹:𝐶𝑊𝑌𝐼) → 𝐻 = (𝑖𝐼 ↦ (𝐹𝑖)))
3 imaeq2 6074 . . 3 (𝑖 = 𝑌 → (𝐹𝑖) = (𝐹𝑌))
43adantl 481 . 2 (((𝐹:𝐶𝑊𝑌𝐼) ∧ 𝑖 = 𝑌) → (𝐹𝑖) = (𝐹𝑌))
5 simpr 484 . 2 ((𝐹:𝐶𝑊𝑌𝐼) → 𝑌𝐼)
6 id 22 . . . . 5 (𝐹:𝐶𝑊𝐹:𝐶𝑊)
7 isubgr3stgr.c . . . . . . 7 𝐶 = (𝐺 ClNeighbVtx 𝑋)
87ovexi 7465 . . . . . 6 𝐶 ∈ V
98a1i 11 . . . . 5 (𝐹:𝐶𝑊𝐶 ∈ V)
106, 9fexd 7247 . . . 4 (𝐹:𝐶𝑊𝐹 ∈ V)
1110adantr 480 . . 3 ((𝐹:𝐶𝑊𝑌𝐼) → 𝐹 ∈ V)
1211imaexd 7938 . 2 ((𝐹:𝐶𝑊𝑌𝐼) → (𝐹𝑌) ∈ V)
132, 4, 5, 12fvmptd 7023 1 ((𝐹:𝐶𝑊𝑌𝐼) → (𝐻𝑌) = (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cmpt 5225  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  0cn0 12526  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349   ClNeighbVtx cclnbgr 47805   ISubGr cisubgr 47846  StarGrcstgr 47918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434
This theorem is referenced by:  isubgr3stgrlem8  47940  isubgr3stgrlem9  47941
  Copyright terms: Public domain W3C validator