| Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgr3stgrlem5 | Structured version Visualization version GIF version | ||
| Description: Lemma 5 for isubgr3stgr 47942. (Contributed by AV, 24-Sep-2025.) |
| 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)) |
| Ref | Expression |
|---|---|
| isubgr3stgrlem5 | ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌)) |
| 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 6074 | . . 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 7465 | . . . . . 6 ⊢ 𝐶 ∈ V |
| 9 | 8 | a1i 11 | . . . . 5 ⊢ (𝐹:𝐶⟶𝑊 → 𝐶 ∈ V) |
| 10 | 6, 9 | fexd 7247 | . . . 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 7023 | 1 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑌 ∈ 𝐼) → (𝐻‘𝑌) = (𝐹 “ 𝑌)) |
| 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 ℕ0cn0 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 |