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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgr3stgrlem9 | Structured version Visualization version GIF version | ||
| Description: Lemma 9 for isubgr3stgr 47942. (Contributed by AV, 29-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 |
|---|---|
| isubgr3stgrlem9 | ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐻:𝐼–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ 𝐼 (𝐹 “ 𝑒) = (𝐻‘𝑒))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isubgr3stgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | isubgr3stgr.u | . . 3 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) | |
| 3 | isubgr3stgr.c | . . 3 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋) | |
| 4 | isubgr3stgr.n | . . 3 ⊢ 𝑁 ∈ ℕ0 | |
| 5 | isubgr3stgr.s | . . 3 ⊢ 𝑆 = (StarGr‘𝑁) | |
| 6 | isubgr3stgr.w | . . 3 ⊢ 𝑊 = (Vtx‘𝑆) | |
| 7 | isubgr3stgr.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
| 8 | isubgr3stgr.i | . . 3 ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶)) | |
| 9 | isubgr3stgr.h | . . 3 ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖)) | |
| 10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isubgr3stgrlem8 47940 | . 2 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐻:𝐼–1-1-onto→(Edg‘(StarGr‘𝑁))) |
| 11 | f1of 6848 | . . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶⟶𝑊) | |
| 12 | 11 | ad2antrl 728 | . . . 4 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐹:𝐶⟶𝑊) |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | isubgr3stgrlem5 47937 | . . . . 5 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑒 ∈ 𝐼) → (𝐻‘𝑒) = (𝐹 “ 𝑒)) |
| 14 | 13 | eqcomd 2743 | . . . 4 ⊢ ((𝐹:𝐶⟶𝑊 ∧ 𝑒 ∈ 𝐼) → (𝐹 “ 𝑒) = (𝐻‘𝑒)) |
| 15 | 12, 14 | sylan 580 | . . 3 ⊢ (((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) ∧ 𝑒 ∈ 𝐼) → (𝐹 “ 𝑒) = (𝐻‘𝑒)) |
| 16 | 15 | ralrimiva 3146 | . 2 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → ∀𝑒 ∈ 𝐼 (𝐹 “ 𝑒) = (𝐻‘𝑒)) |
| 17 | 10, 16 | jca 511 | 1 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → (𝐻:𝐼–1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒 ∈ 𝐼 (𝐹 “ 𝑒) = (𝐻‘𝑒))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 ∀wral 3061 {cpr 4628 ↦ cmpt 5225 “ cima 5688 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕ0cn0 12526 ♯chash 14369 Vtxcvtx 29013 Edgcedg 29064 USGraphcusgr 29166 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-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3047 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-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-hash 14370 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-edgf 29004 df-vtx 29015 df-iedg 29016 df-edg 29065 df-uhgr 29075 df-upgr 29099 df-umgr 29100 df-uspgr 29167 df-usgr 29168 df-nbgr 29350 df-clnbgr 47806 df-isubgr 47847 df-stgr 47919 |
| This theorem is referenced by: isubgr3stgr 47942 |
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