Theorem isubgr3stgrlem5 48763

Description: Lemma 5 for isubgr3stgr 48768. (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 6058	. . 3 ⊢ (𝑖 = 𝑌 → (𝐹 “ 𝑖) = (𝐹 “ 𝑌))
4	3	adantl 486	. 2 ⊢ (((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) ∧ 𝑖 = 𝑌) → (𝐹 “ 𝑖) = (𝐹 “ 𝑌))
5		simpr 489	. 2 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ 𝐼)
6		id 23	. . . . 5 ⊢ (𝐹:𝐶⟶𝑊 → 𝐹:𝐶⟶𝑊)
7		isubgr3stgr.c	. . . . . . 7 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
8	7	ovexi 7444	. . . . . 6 ⊢ 𝐶 ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝐹:𝐶⟶𝑊 → 𝐶 ∈ V)
10	6, 9	fexd 7225	. . . 4 ⊢ (𝐹:𝐶⟶𝑊 → 𝐹 ∈ V)
11	10	adantr 485	. . 3 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → 𝐹 ∈ V)
12	11	imaexd 7909	. 2 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐹 “ 𝑌) ∈ V)
13	2, 4, 5, 12	fvmptd 6997	1 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ↦ cmpt 5192   “ cima 5664  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℕ₀cn0 12508  Vtxcvtx 29355  Edgcedg 29406   NeighbVtx cnbgr 29691   ClNeighbVtx cclnbgr 48611   ISubGr cisubgr 48653  StarGrcstgr 48744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413

This theorem is used by:  isubgr3stgrlem8  48766  isubgr3stgrlem9  48767