Theorem isubgr3stgrlem5 47872

Description: Lemma 5 for isubgr3stgr 47877. (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 6075	. . 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 7464	. . . . . 6 ⊢ 𝐶 ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝐹:𝐶⟶𝑊 → 𝐶 ∈ V)
10	6, 9	fexd 7246	. . . 4 ⊢ (𝐹:𝐶⟶𝑊 → 𝐹 ∈ V)
11	10	adantr 480	. . 3 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → 𝐹 ∈ V)
12	11	imaexd 7938	. 2 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐹 “ 𝑌) ∈ V)
13	2, 4, 5, 12	fvmptd 7022	1 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  Vcvv 3477   ↦ cmpt 5230   “ cima 5691  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  ℕ₀cn0 12523  Vtxcvtx 29027  Edgcedg 29078   NeighbVtx cnbgr 29363   ClNeighbVtx cclnbgr 47742   ISubGr cisubgr 47783  StarGrcstgr 47853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433

This theorem is referenced by:  isubgr3stgrlem8  47875  isubgr3stgrlem9  47876