Theorem isubgr3stgrlem5 48359

Description: Lemma 5 for isubgr3stgr 48364. (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 6025	. . 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 7404	. . . . . 6 ⊢ 𝐶 ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝐹:𝐶⟶𝑊 → 𝐶 ∈ V)
10	6, 9	fexd 7185	. . . 4 ⊢ (𝐹:𝐶⟶𝑊 → 𝐹 ∈ V)
11	10	adantr 480	. . 3 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → 𝐹 ∈ V)
12	11	imaexd 7870	. 2 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐹 “ 𝑌) ∈ V)
13	2, 4, 5, 12	fvmptd 6959	1 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ↦ cmpt 5181   “ cima 5637  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370  ℕ₀cn0 12415  Vtxcvtx 29087  Edgcedg 29138   NeighbVtx cnbgr 29423   ClNeighbVtx cclnbgr 48207   ISubGr cisubgr 48249  StarGrcstgr 48340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373

This theorem is referenced by:  isubgr3stgrlem8  48362  isubgr3stgrlem9  48363