Theorem isubgr3stgrlem5 47964

Description: Lemma 5 for isubgr3stgr 47969. (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 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))

Assertion

Ref	Expression
isubgr3stgrlem5	⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌))

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

Step	Hyp	Ref	Expression
1		isubgr3stgr.h	. . 3 ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))
2	1	a1i 11	. 2 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖)))
3		imaeq2 6007	. . 3 ⊢ (𝑖 = 𝑌 → (𝐹 “ 𝑖) = (𝐹 “ 𝑌))
4	3	adantl 481	. 2 ⊢ (((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) ∧ 𝑖 = 𝑌) → (𝐹 “ 𝑖) = (𝐹 “ 𝑌))
5		simpr 484	. 2 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ 𝐼)
6		id 22	. . . . 5 ⊢ (𝐹:𝐶⟶𝑊 → 𝐹:𝐶⟶𝑊)
7		isubgr3stgr.c	. . . . . . 7 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
8	7	ovexi 7383	. . . . . 6 ⊢ 𝐶 ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝐹:𝐶⟶𝑊 → 𝐶 ∈ V)
10	6, 9	fexd 7163	. . . 4 ⊢ (𝐹:𝐶⟶𝑊 → 𝐹 ∈ V)
11	10	adantr 480	. . 3 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → 𝐹 ∈ V)
12	11	imaexd 7849	. 2 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐹 “ 𝑌) ∈ V)
13	2, 4, 5, 12	fvmptd 6937	1 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ↦ cmpt 5173   “ cima 5622  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ℕ₀cn0 12384  Vtxcvtx 28941  Edgcedg 28992   NeighbVtx cnbgr 29277   ClNeighbVtx cclnbgr 47812   ISubGr cisubgr 47854  StarGrcstgr 47945

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352

This theorem is referenced by:  isubgr3stgrlem8  47967  isubgr3stgrlem9  47968