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Theorem isubgr3stgrlem7 47874
Description: Lemma 7 for isubgr3stgr 47877. (Contributed by AV, 29-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
isubgr3stgrlem7 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ 𝐽 ∈ (Edg‘(StarGr‘𝑁))) → (𝐹𝐽) ∈ 𝐼)
Distinct variable groups:   𝐶,𝑖   𝑖,𝐹   𝑖,𝐼   𝑖,𝑊   𝑖,𝐸   𝑖,𝐺   𝑖,𝑁   𝑈,𝑖   𝑖,𝑉   𝑖,𝑋
Allowed substitution hints:   𝑆(𝑖)   𝐻(𝑖)   𝐽(𝑖)

Proof of Theorem isubgr3stgrlem7
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isubgr3stgr.n . . . 4 𝑁 ∈ ℕ0
2 stgredgel 47859 . . . 4 (𝑁 ∈ ℕ0 → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) ↔ (𝐽 ⊆ (0...𝑁) ∧ ∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦})))
31, 2mp1i 13 . . 3 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) ↔ (𝐽 ⊆ (0...𝑁) ∧ ∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦})))
4 c0ex 11252 . . . . . . . . . . . . 13 0 ∈ V
54a1i 11 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 0 ∈ V)
6 prssg 4823 . . . . . . . . . . . 12 ((0 ∈ V ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ {0, 𝑦} ⊆ (0...𝑁)))
75, 6sylan 580 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ {0, 𝑦} ⊆ (0...𝑁)))
8 f1ocnv 6860 . . . . . . . . . . . . . . . . 17 (𝐹:𝐶1-1-onto𝑊𝐹:𝑊1-1-onto𝐶)
9 f1ofn 6849 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑊1-1-onto𝐶𝐹 Fn 𝑊)
10 isubgr3stgr.w . . . . . . . . . . . . . . . . . . . 20 𝑊 = (Vtx‘𝑆)
11 isubgr3stgr.s . . . . . . . . . . . . . . . . . . . . 21 𝑆 = (StarGr‘𝑁)
1211fveq2i 6909 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
13 stgrvtx 47856 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
141, 13ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘(StarGr‘𝑁)) = (0...𝑁)
1510, 12, 143eqtri 2766 . . . . . . . . . . . . . . . . . . 19 𝑊 = (0...𝑁)
1615fneq2i 6666 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑊𝐹 Fn (0...𝑁))
179, 16sylib 218 . . . . . . . . . . . . . . . . 17 (𝐹:𝑊1-1-onto𝐶𝐹 Fn (0...𝑁))
188, 17syl 17 . . . . . . . . . . . . . . . 16 (𝐹:𝐶1-1-onto𝑊𝐹 Fn (0...𝑁))
1918ad2antrl 728 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐹 Fn (0...𝑁))
2019adantr 480 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝐹 Fn (0...𝑁))
2120anim1i 615 . . . . . . . . . . . . 13 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))) → (𝐹 Fn (0...𝑁) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
22 3anass 1094 . . . . . . . . . . . . 13 ((𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) ↔ (𝐹 Fn (0...𝑁) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
2321, 22sylibr 234 . . . . . . . . . . . 12 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))) → (𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)))
2423ex 412 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) → (𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
257, 24sylbird 260 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ({0, 𝑦} ⊆ (0...𝑁) → (𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁))))
2625imp 406 . . . . . . . . 9 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → (𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)))
27 fnimapr 6991 . . . . . . . . 9 ((𝐹 Fn (0...𝑁) ∧ 0 ∈ (0...𝑁) ∧ 𝑦 ∈ (0...𝑁)) → (𝐹 “ {0, 𝑦}) = {(𝐹‘0), (𝐹𝑦)})
2826, 27syl 17 . . . . . . . 8 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → (𝐹 “ {0, 𝑦}) = {(𝐹‘0), (𝐹𝑦)})
29 isubgr3stgr.v . . . . . . . . . . . . . . . . . 18 𝑉 = (Vtx‘𝐺)
3029clnbgrvtxel 47753 . . . . . . . . . . . . . . . . 17 (𝑋𝑉𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
3130adantl 481 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → 𝑋 ∈ (𝐺 ClNeighbVtx 𝑋))
32 isubgr3stgr.c . . . . . . . . . . . . . . . 16 𝐶 = (𝐺 ClNeighbVtx 𝑋)
3331, 32eleqtrrdi 2849 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → 𝑋𝐶)
34 simpl 482 . . . . . . . . . . . . . . 15 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → 𝐹:𝐶1-1-onto𝑊)
3533, 34anim12ci 614 . . . . . . . . . . . . . 14 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝐹:𝐶1-1-onto𝑊𝑋𝐶))
36 simprr 773 . . . . . . . . . . . . . 14 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝐹𝑋) = 0)
3735, 36jca 511 . . . . . . . . . . . . 13 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → ((𝐹:𝐶1-1-onto𝑊𝑋𝐶) ∧ (𝐹𝑋) = 0))
3837adantr 480 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((𝐹:𝐶1-1-onto𝑊𝑋𝐶) ∧ (𝐹𝑋) = 0))
39 f1ocnvfv 7297 . . . . . . . . . . . . 13 ((𝐹:𝐶1-1-onto𝑊𝑋𝐶) → ((𝐹𝑋) = 0 → (𝐹‘0) = 𝑋))
4039imp 406 . . . . . . . . . . . 12 (((𝐹:𝐶1-1-onto𝑊𝑋𝐶) ∧ (𝐹𝑋) = 0) → (𝐹‘0) = 𝑋)
4138, 40syl 17 . . . . . . . . . . 11 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐹‘0) = 𝑋)
4230, 32eleqtrrdi 2849 . . . . . . . . . . . . . . 15 (𝑋𝑉𝑋𝐶)
4342ad3antlr 731 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑋𝐶)
44 f1of 6848 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑊1-1-onto𝐶𝐹:𝑊𝐶)
458, 44syl 17 . . . . . . . . . . . . . . . . 17 (𝐹:𝐶1-1-onto𝑊𝐹:𝑊𝐶)
4645ad2antrl 728 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐹:𝑊𝐶)
4746adantr 480 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝐹:𝑊𝐶)
48 fz1ssfz0 13659 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ⊆ (0...𝑁)
4948sseli 3990 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1...𝑁) → 𝑦 ∈ (0...𝑁))
5049, 15eleqtrrdi 2849 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1...𝑁) → 𝑦𝑊)
5150adantl 481 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑦𝑊)
5247, 51ffvelcdmd 7104 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐹𝑦) ∈ 𝐶)
5343, 52jca 511 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝑋𝐶 ∧ (𝐹𝑦) ∈ 𝐶))
5432eleq2i 2830 . . . . . . . . . . . . . . . . 17 ((𝐹𝑦) ∈ 𝐶 ↔ (𝐹𝑦) ∈ (𝐺 ClNeighbVtx 𝑋))
55 usgrupgr 29216 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
5655anim1i 615 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → (𝐺 ∈ UPGraph ∧ 𝑋𝑉))
5756ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐺 ∈ UPGraph ∧ 𝑋𝑉))
5829clnbgrssvtx 47755 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
5932, 58eqsstri 4029 . . . . . . . . . . . . . . . . . . . . 21 𝐶𝑉
6059, 52sselid 3992 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐹𝑦) ∈ 𝑉)
61 df-3an 1088 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ UPGraph ∧ 𝑋𝑉 ∧ (𝐹𝑦) ∈ 𝑉) ↔ ((𝐺 ∈ UPGraph ∧ 𝑋𝑉) ∧ (𝐹𝑦) ∈ 𝑉))
6257, 60, 61sylanbrc 583 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐺 ∈ UPGraph ∧ 𝑋𝑉 ∧ (𝐹𝑦) ∈ 𝑉))
6362ad2antrr 726 . . . . . . . . . . . . . . . . . 18 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ 𝑋𝐶) → (𝐺 ∈ UPGraph ∧ 𝑋𝑉 ∧ (𝐹𝑦) ∈ 𝑉))
64 isubgr3stgr.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edg‘𝐺)
6529, 64clnbupgrel 47758 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ UPGraph ∧ 𝑋𝑉 ∧ (𝐹𝑦) ∈ 𝑉) → ((𝐹𝑦) ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝐹𝑦) = 𝑋 ∨ {(𝐹𝑦), 𝑋} ∈ 𝐸)))
6663, 65syl 17 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ 𝑋𝐶) → ((𝐹𝑦) ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝐹𝑦) = 𝑋 ∨ {(𝐹𝑦), 𝑋} ∈ 𝐸)))
6754, 66bitrid 283 . . . . . . . . . . . . . . . 16 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ 𝑋𝐶) → ((𝐹𝑦) ∈ 𝐶 ↔ ((𝐹𝑦) = 𝑋 ∨ {(𝐹𝑦), 𝑋} ∈ 𝐸)))
68 eqeq2 2746 . . . . . . . . . . . . . . . . . . . 20 ((𝐹‘0) = 𝑋 → ((𝐹𝑦) = (𝐹‘0) ↔ (𝐹𝑦) = 𝑋))
6968adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) → ((𝐹𝑦) = (𝐹‘0) ↔ (𝐹𝑦) = 𝑋))
70 f1of1 6847 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝑊1-1-onto𝐶𝐹:𝑊1-1𝐶)
718, 70syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝐶1-1-onto𝑊𝐹:𝑊1-1𝐶)
7271ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐹:𝑊1-1𝐶)
73 0elfz 13660 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
741, 73ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ (0...𝑁)
7574, 15eleqtrri 2837 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ 𝑊
7650, 75jctir 520 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (1...𝑁) → (𝑦𝑊 ∧ 0 ∈ 𝑊))
77 f1veqaeq 7276 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝑊1-1𝐶 ∧ (𝑦𝑊 ∧ 0 ∈ 𝑊)) → ((𝐹𝑦) = (𝐹‘0) → 𝑦 = 0))
7872, 76, 77syl2an 596 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((𝐹𝑦) = (𝐹‘0) → 𝑦 = 0))
79 elfznn 13589 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
80 nnne0 12297 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
81 eqneqall 2948 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 0 → (𝑦 ≠ 0 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
8280, 81syl5com 31 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℕ → (𝑦 = 0 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
8379, 82syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (1...𝑁) → (𝑦 = 0 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
8483adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝑦 = 0 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
8578, 84syld 47 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ((𝐹𝑦) = (𝐹‘0) → {𝑋, (𝐹𝑦)} ∈ 𝐸))
8685adantr 480 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) → ((𝐹𝑦) = (𝐹‘0) → {𝑋, (𝐹𝑦)} ∈ 𝐸))
8769, 86sylbird 260 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) → ((𝐹𝑦) = 𝑋 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
8887adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ 𝑋𝐶) → ((𝐹𝑦) = 𝑋 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
89 prcom 4736 . . . . . . . . . . . . . . . . . . . 20 {(𝐹𝑦), 𝑋} = {𝑋, (𝐹𝑦)}
9089eleq1i 2829 . . . . . . . . . . . . . . . . . . 19 ({(𝐹𝑦), 𝑋} ∈ 𝐸 ↔ {𝑋, (𝐹𝑦)} ∈ 𝐸)
9190biimpi 216 . . . . . . . . . . . . . . . . . 18 ({(𝐹𝑦), 𝑋} ∈ 𝐸 → {𝑋, (𝐹𝑦)} ∈ 𝐸)
9291a1i 11 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ 𝑋𝐶) → ({(𝐹𝑦), 𝑋} ∈ 𝐸 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
9388, 92jaod 859 . . . . . . . . . . . . . . . 16 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ 𝑋𝐶) → (((𝐹𝑦) = 𝑋 ∨ {(𝐹𝑦), 𝑋} ∈ 𝐸) → {𝑋, (𝐹𝑦)} ∈ 𝐸))
9467, 93sylbid 240 . . . . . . . . . . . . . . 15 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ 𝑋𝐶) → ((𝐹𝑦) ∈ 𝐶 → {𝑋, (𝐹𝑦)} ∈ 𝐸))
9594impr 454 . . . . . . . . . . . . . 14 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ (𝑋𝐶 ∧ (𝐹𝑦) ∈ 𝐶)) → {𝑋, (𝐹𝑦)} ∈ 𝐸)
96 prssi 4825 . . . . . . . . . . . . . . 15 ((𝑋𝐶 ∧ (𝐹𝑦) ∈ 𝐶) → {𝑋, (𝐹𝑦)} ⊆ 𝐶)
9796adantl 481 . . . . . . . . . . . . . 14 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ (𝑋𝐶 ∧ (𝐹𝑦) ∈ 𝐶)) → {𝑋, (𝐹𝑦)} ⊆ 𝐶)
9895, 97jca 511 . . . . . . . . . . . . 13 ((((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) ∧ (𝑋𝐶 ∧ (𝐹𝑦) ∈ 𝐶)) → ({𝑋, (𝐹𝑦)} ∈ 𝐸 ∧ {𝑋, (𝐹𝑦)} ⊆ 𝐶))
9953, 98mpidan 689 . . . . . . . . . . . 12 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) → ({𝑋, (𝐹𝑦)} ∈ 𝐸 ∧ {𝑋, (𝐹𝑦)} ⊆ 𝐶))
100 preq1 4737 . . . . . . . . . . . . . . 15 ((𝐹‘0) = 𝑋 → {(𝐹‘0), (𝐹𝑦)} = {𝑋, (𝐹𝑦)})
101100eleq1d 2823 . . . . . . . . . . . . . 14 ((𝐹‘0) = 𝑋 → ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ↔ {𝑋, (𝐹𝑦)} ∈ 𝐸))
102100sseq1d 4026 . . . . . . . . . . . . . 14 ((𝐹‘0) = 𝑋 → ({(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶 ↔ {𝑋, (𝐹𝑦)} ⊆ 𝐶))
103101, 102anbi12d 632 . . . . . . . . . . . . 13 ((𝐹‘0) = 𝑋 → (({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶) ↔ ({𝑋, (𝐹𝑦)} ∈ 𝐸 ∧ {𝑋, (𝐹𝑦)} ⊆ 𝐶)))
104103adantl 481 . . . . . . . . . . . 12 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) → (({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶) ↔ ({𝑋, (𝐹𝑦)} ∈ 𝐸 ∧ {𝑋, (𝐹𝑦)} ⊆ 𝐶)))
10599, 104mpbird 257 . . . . . . . . . . 11 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝐹‘0) = 𝑋) → ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶))
10641, 105mpdan 687 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶))
107106adantr 480 . . . . . . . . 9 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶))
108 usgruhgr 29217 . . . . . . . . . . 11 (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
109108ad3antrrr 730 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝐺 ∈ UHGraph)
11059a1i 11 . . . . . . . . . 10 ({0, 𝑦} ⊆ (0...𝑁) → 𝐶𝑉)
111 eqid 2734 . . . . . . . . . . 11 (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
112 isubgr3stgr.i . . . . . . . . . . 11 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
11329, 64, 111, 112isubgredg 47789 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝐶𝑉) → ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐼 ↔ ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶)))
114109, 110, 113syl2an 596 . . . . . . . . 9 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐼 ↔ ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹𝑦)} ⊆ 𝐶)))
115107, 114mpbird 257 . . . . . . . 8 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → {(𝐹‘0), (𝐹𝑦)} ∈ 𝐼)
11628, 115eqeltrd 2838 . . . . . . 7 (((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) ∧ {0, 𝑦} ⊆ (0...𝑁)) → (𝐹 “ {0, 𝑦}) ∈ 𝐼)
117116ex 412 . . . . . 6 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → ({0, 𝑦} ⊆ (0...𝑁) → (𝐹 “ {0, 𝑦}) ∈ 𝐼))
118 sseq1 4020 . . . . . . 7 (𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) ↔ {0, 𝑦} ⊆ (0...𝑁)))
119 imaeq2 6075 . . . . . . . 8 (𝐽 = {0, 𝑦} → (𝐹𝐽) = (𝐹 “ {0, 𝑦}))
120119eleq1d 2823 . . . . . . 7 (𝐽 = {0, 𝑦} → ((𝐹𝐽) ∈ 𝐼 ↔ (𝐹 “ {0, 𝑦}) ∈ 𝐼))
121118, 120imbi12d 344 . . . . . 6 (𝐽 = {0, 𝑦} → ((𝐽 ⊆ (0...𝑁) → (𝐹𝐽) ∈ 𝐼) ↔ ({0, 𝑦} ⊆ (0...𝑁) → (𝐹 “ {0, 𝑦}) ∈ 𝐼)))
122117, 121syl5ibrcom 247 . . . . 5 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) → (𝐹𝐽) ∈ 𝐼)))
123122rexlimdva 3152 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) → (𝐹𝐽) ∈ 𝐼)))
124123impcomd 411 . . 3 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → ((𝐽 ⊆ (0...𝑁) ∧ ∃𝑦 ∈ (1...𝑁)𝐽 = {0, 𝑦}) → (𝐹𝐽) ∈ 𝐼))
1253, 124sylbid 240 . 2 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝐽 ∈ (Edg‘(StarGr‘𝑁)) → (𝐹𝐽) ∈ 𝐼))
1261253impia 1116 1 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ 𝐽 ∈ (Edg‘(StarGr‘𝑁))) → (𝐹𝐽) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  wss 3962  {cpr 4632  cmpt 5230  ccnv 5687  cima 5691   Fn wfn 6557  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153  cn 12263  0cn0 12523  ...cfz 13543  Vtxcvtx 29027  Edgcedg 29078  UHGraphcuhgr 29087  UPGraphcupgr 29111  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-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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