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Theorem isubgr3stgrlem6 47873
Description: Lemma 6 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 = (𝑖𝐼 ↦ (𝐹𝑖))
Assertion
Ref Expression
isubgr3stgrlem6 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐻:𝐼⟶(Edg‘(StarGr‘𝑁)))
Distinct variable groups:   𝐶,𝑖   𝑖,𝐹   𝑖,𝐼   𝑖,𝑊   𝑖,𝐸,𝑥,𝑦   𝑖,𝐺   𝑖,𝑁   𝑈,𝑖,𝑥,𝑦   𝑖,𝑉   𝑖,𝑋
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑖)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦,𝑖)   𝐼(𝑥,𝑦)   𝑁(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem isubgr3stgrlem6
Dummy variables 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgruhgr 29217 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
21adantr 480 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → 𝐺 ∈ UHGraph)
32adantr 480 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐺 ∈ UHGraph)
4 isubgr3stgr.c . . . . . 6 𝐶 = (𝐺 ClNeighbVtx 𝑋)
5 isubgr3stgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
65clnbgrssvtx 47755 . . . . . 6 (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
74, 6eqsstri 4029 . . . . 5 𝐶𝑉
87a1i 11 . . . 4 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → 𝐶𝑉)
9 isubgr3stgr.e . . . . 5 𝐸 = (Edg‘𝐺)
10 eqid 2734 . . . . 5 (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
11 isubgr3stgr.i . . . . 5 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
125, 9, 10, 11isubgredg 47789 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐶𝑉) → (𝑖𝐼 ↔ (𝑖𝐸𝑖𝐶)))
133, 8, 12syl2an 596 . . 3 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝑖𝐼 ↔ (𝑖𝐸𝑖𝐶)))
14 f1of 6848 . . . . . . . 8 (𝐹:𝐶1-1-onto𝑊𝐹:𝐶𝑊)
15 isubgr3stgr.w . . . . . . . . . . 11 𝑊 = (Vtx‘𝑆)
16 isubgr3stgr.s . . . . . . . . . . . 12 𝑆 = (StarGr‘𝑁)
1716fveq2i 6909 . . . . . . . . . . 11 (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
18 isubgr3stgr.n . . . . . . . . . . . 12 𝑁 ∈ ℕ0
19 stgrvtx 47856 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
2018, 19ax-mp 5 . . . . . . . . . . 11 (Vtx‘(StarGr‘𝑁)) = (0...𝑁)
2115, 17, 203eqtri 2766 . . . . . . . . . 10 𝑊 = (0...𝑁)
2221eqimssi 4055 . . . . . . . . 9 𝑊 ⊆ (0...𝑁)
2322a1i 11 . . . . . . . 8 (𝐹:𝐶1-1-onto𝑊𝑊 ⊆ (0...𝑁))
2414, 23fssd 6753 . . . . . . 7 (𝐹:𝐶1-1-onto𝑊𝐹:𝐶⟶(0...𝑁))
2524ad2antrl 728 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐹:𝐶⟶(0...𝑁))
2625adantr 480 . . . . 5 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → 𝐹:𝐶⟶(0...𝑁))
2726fimassd 6757 . . . 4 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → (𝐹𝑖) ⊆ (0...𝑁))
28 simplll 775 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐺 ∈ USGraph)
29 simpl 482 . . . . . 6 ((𝑖𝐸𝑖𝐶) → 𝑖𝐸)
305, 9usgredg 29230 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑖𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑖 = {𝑎, 𝑏}))
3128, 29, 30syl2an 596 . . . . 5 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑖 = {𝑎, 𝑏}))
32 vex 3481 . . . . . . . . . . . . . . . . 17 𝑎 ∈ V
33 vex 3481 . . . . . . . . . . . . . . . . 17 𝑏 ∈ V
3432, 33prss 4824 . . . . . . . . . . . . . . . 16 ((𝑎𝐶𝑏𝐶) ↔ {𝑎, 𝑏} ⊆ 𝐶)
35 elclnbgrelnbgr 47749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑎 ∈ (𝐺 ClNeighbVtx 𝑋) ∧ 𝑎𝑋) → 𝑎 ∈ (𝐺 NeighbVtx 𝑋))
3635expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝑋 → (𝑎 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑎 ∈ (𝐺 NeighbVtx 𝑋)))
374eleq2i 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝐶𝑎 ∈ (𝐺 ClNeighbVtx 𝑋))
38 isubgr3stgr.u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑈 = (𝐺 NeighbVtx 𝑋)
3938eleq2i 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝑈𝑎 ∈ (𝐺 NeighbVtx 𝑋))
4036, 37, 393imtr4g 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎𝑋 → (𝑎𝐶𝑎𝑈))
41 elclnbgrelnbgr 47749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ (𝐺 ClNeighbVtx 𝑋) ∧ 𝑏𝑋) → 𝑏 ∈ (𝐺 NeighbVtx 𝑋))
4241expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏𝑋 → (𝑏 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑏 ∈ (𝐺 NeighbVtx 𝑋)))
434eleq2i 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏𝐶𝑏 ∈ (𝐺 ClNeighbVtx 𝑋))
4438eleq2i 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏𝑈𝑏 ∈ (𝐺 NeighbVtx 𝑋))
4542, 43, 443imtr4g 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏𝑋 → (𝑏𝐶𝑏𝑈))
4640, 45im2anan9r 621 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏𝑋𝑎𝑋) → ((𝑎𝐶𝑏𝐶) → (𝑎𝑈𝑏𝑈)))
4746imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑏𝑋𝑎𝑋) ∧ (𝑎𝐶𝑏𝐶)) → (𝑎𝑈𝑏𝑈))
48473adant3 1131 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑏𝑋𝑎𝑋) ∧ (𝑎𝐶𝑏𝐶) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → (𝑎𝑈𝑏𝑈))
49 preq1 4737 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑎 → {𝑥, 𝑦} = {𝑎, 𝑦})
50 eqidd 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑎𝐸 = 𝐸)
5149, 50neleq12d 3048 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑎 → ({𝑥, 𝑦} ∉ 𝐸 ↔ {𝑎, 𝑦} ∉ 𝐸))
52 preq2 4738 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑏 → {𝑎, 𝑦} = {𝑎, 𝑏})
53 eqidd 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑏𝐸 = 𝐸)
5452, 53neleq12d 3048 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑏 → ({𝑎, 𝑦} ∉ 𝐸 ↔ {𝑎, 𝑏} ∉ 𝐸))
5551, 54rspc2v 3632 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑈𝑏𝑈) → (∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸 → {𝑎, 𝑏} ∉ 𝐸))
5648, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑏𝑋𝑎𝑋) ∧ (𝑎𝐶𝑏𝐶) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → (∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸 → {𝑎, 𝑏} ∉ 𝐸))
57 pm2.24nel 3056 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝑎, 𝑏} ∈ 𝐸 → ({𝑎, 𝑏} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸) → ({𝑎, 𝑏} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
59583ad2ant3 1134 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑏𝑋𝑎𝑋) ∧ (𝑎𝐶𝑏𝐶) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → ({𝑎, 𝑏} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
6056, 59syld 47 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑏𝑋𝑎𝑋) ∧ (𝑎𝐶𝑏𝐶) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → (∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
61603exp 1118 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏𝑋𝑎𝑋) → ((𝑎𝐶𝑏𝐶) → ((𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸) → (∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
6261com24 95 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑋𝑎𝑋) → (∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸 → ((𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸) → ((𝑎𝐶𝑏𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
6362adantld 490 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏𝑋𝑎𝑋) → (((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸) → ((𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸) → ((𝑎𝐶𝑏𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
6463adantld 490 . . . . . . . . . . . . . . . . . . . 20 ((𝑏𝑋𝑎𝑋) → (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) → ((𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸) → ((𝑎𝐶𝑏𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
6564adantrd 491 . . . . . . . . . . . . . . . . . . 19 ((𝑏𝑋𝑎𝑋) → ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → ((𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸) → ((𝑎𝐶𝑏𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
6665imp4c 423 . . . . . . . . . . . . . . . . . 18 ((𝑏𝑋𝑎𝑋) → ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
67 simpl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑏 = 𝑋)
68 simpllr 776 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶)) → (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0))
6968adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0))
70 simplrl 777 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶)) → 𝑎𝑏)
7170necomd 2993 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶)) → 𝑏𝑎)
7271adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑏𝑎)
73 simprrr 782 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑏𝐶)
74 simprrl 781 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑎𝐶)
755, 38, 4, 18, 16, 15, 9isubgr3stgrlem4 47871 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑋 ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑏𝑎𝑏𝐶𝑎𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑏, 𝑎}) = {0, 𝑧})
7667, 69, 72, 73, 74, 75syl113anc 1381 . . . . . . . . . . . . . . . . . . . 20 ((𝑏 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑏, 𝑎}) = {0, 𝑧})
77 prcom 4736 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑎, 𝑏} = {𝑏, 𝑎}
7877imaeq2i 6077 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 “ {𝑎, 𝑏}) = (𝐹 “ {𝑏, 𝑎})
7978eqeq1i 2739 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 “ {𝑎, 𝑏}) = {0, 𝑧} ↔ (𝐹 “ {𝑏, 𝑎}) = {0, 𝑧})
8079rexbii 3091 . . . . . . . . . . . . . . . . . . . 20 (∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧} ↔ ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑏, 𝑎}) = {0, 𝑧})
8176, 80sylibr 234 . . . . . . . . . . . . . . . . . . 19 ((𝑏 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})
8281ex 412 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑋 → ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
83 simpl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑎 = 𝑋)
8468adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0))
8570adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑎𝑏)
86 simprrl 781 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑎𝐶)
87 simprrr 782 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → 𝑏𝐶)
885, 38, 4, 18, 16, 15, 9isubgr3stgrlem4 47871 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = 𝑋 ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑎𝑏𝑎𝐶𝑏𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})
8983, 84, 85, 86, 87, 88syl113anc 1381 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝑋 ∧ (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶))) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})
9089ex 412 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑋 → ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
9166, 82, 90pm2.61iine 3029 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) ∧ (𝑎𝐶𝑏𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})
9291ex 412 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → ((𝑎𝐶𝑏𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
9334, 92biimtrrid 243 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → ({𝑎, 𝑏} ⊆ 𝐶 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
9493exp32 420 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝑎𝑏 → ({𝑎, 𝑏} ∈ 𝐸 → ({𝑎, 𝑏} ⊆ 𝐶 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
9594adantrd 491 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → ((𝑎𝑏𝑖 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ∈ 𝐸 → ({𝑎, 𝑏} ⊆ 𝐶 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
9695imp 406 . . . . . . . . . . . 12 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏𝑖 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → ({𝑎, 𝑏} ⊆ 𝐶 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})))
9796com23 86 . . . . . . . . . . 11 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏𝑖 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ⊆ 𝐶 → ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})))
98 sseq1 4020 . . . . . . . . . . . . . 14 (𝑖 = {𝑎, 𝑏} → (𝑖𝐶 ↔ {𝑎, 𝑏} ⊆ 𝐶))
99 eleq1 2826 . . . . . . . . . . . . . . 15 (𝑖 = {𝑎, 𝑏} → (𝑖𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
100 imaeq2 6075 . . . . . . . . . . . . . . . . 17 (𝑖 = {𝑎, 𝑏} → (𝐹𝑖) = (𝐹 “ {𝑎, 𝑏}))
101100eqeq1d 2736 . . . . . . . . . . . . . . . 16 (𝑖 = {𝑎, 𝑏} → ((𝐹𝑖) = {0, 𝑧} ↔ (𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
102101rexbidv 3176 . . . . . . . . . . . . . . 15 (𝑖 = {𝑎, 𝑏} → (∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧} ↔ ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
10399, 102imbi12d 344 . . . . . . . . . . . . . 14 (𝑖 = {𝑎, 𝑏} → ((𝑖𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧}) ↔ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})))
10498, 103imbi12d 344 . . . . . . . . . . . . 13 (𝑖 = {𝑎, 𝑏} → ((𝑖𝐶 → (𝑖𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧})) ↔ ({𝑎, 𝑏} ⊆ 𝐶 → ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
105104adantl 481 . . . . . . . . . . . 12 ((𝑎𝑏𝑖 = {𝑎, 𝑏}) → ((𝑖𝐶 → (𝑖𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧})) ↔ ({𝑎, 𝑏} ⊆ 𝐶 → ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
106105adantl 481 . . . . . . . . . . 11 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏𝑖 = {𝑎, 𝑏})) → ((𝑖𝐶 → (𝑖𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧})) ↔ ({𝑎, 𝑏} ⊆ 𝐶 → ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))))
10797, 106mpbird 257 . . . . . . . . . 10 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑎𝑏𝑖 = {𝑎, 𝑏})) → (𝑖𝐶 → (𝑖𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧})))
108107ex 412 . . . . . . . . 9 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → ((𝑎𝑏𝑖 = {𝑎, 𝑏}) → (𝑖𝐶 → (𝑖𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧}))))
109108com24 95 . . . . . . . 8 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝑖𝐸 → (𝑖𝐶 → ((𝑎𝑏𝑖 = {𝑎, 𝑏}) → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧}))))
110109imp32 418 . . . . . . 7 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → ((𝑎𝑏𝑖 = {𝑎, 𝑏}) → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧}))
111110a1d 25 . . . . . 6 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → ((𝑎𝑉𝑏𝑉) → ((𝑎𝑏𝑖 = {𝑎, 𝑏}) → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧})))
112111rexlimdvv 3209 . . . . 5 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑖 = {𝑎, 𝑏}) → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧}))
11331, 112mpd 15 . . . 4 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧})
114 stgredgel 47859 . . . . 5 (𝑁 ∈ ℕ0 → ((𝐹𝑖) ∈ (Edg‘(StarGr‘𝑁)) ↔ ((𝐹𝑖) ⊆ (0...𝑁) ∧ ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧})))
11518, 114ax-mp 5 . . . 4 ((𝐹𝑖) ∈ (Edg‘(StarGr‘𝑁)) ↔ ((𝐹𝑖) ⊆ (0...𝑁) ∧ ∃𝑧 ∈ (1...𝑁)(𝐹𝑖) = {0, 𝑧}))
11627, 113, 115sylanbrc 583 . . 3 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ (𝑖𝐸𝑖𝐶)) → (𝐹𝑖) ∈ (Edg‘(StarGr‘𝑁)))
11713, 116sylbida 592 . 2 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑖𝐼) → (𝐹𝑖) ∈ (Edg‘(StarGr‘𝑁)))
118 isubgr3stgr.h . 2 𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))
119117, 118fmptd 7133 1 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐻:𝐼⟶(Edg‘(StarGr‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wnel 3043  wral 3058  wrex 3067  wss 3962  {cpr 4632  cmpt 5230  cima 5691  wf 6558  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153  0cn0 12523  ...cfz 13543  chash 14365  Vtxcvtx 29027  Edgcedg 29078  UHGraphcuhgr 29087  USGraphcusgr 29180   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-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3044  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-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  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-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  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-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-hash 14366  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-edgf 29018  df-vtx 29029  df-iedg 29030  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-umgr 29114  df-uspgr 29181  df-usgr 29182  df-nbgr 29364  df-clnbgr 47743  df-isubgr 47784  df-stgr 47854
This theorem is referenced by:  isubgr3stgrlem8  47875
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