Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isubgr3stgrlem7 Structured version   Visualization version   GIF version

Theorem isubgr3stgrlem7 47939
Description: Lemma 7 for isubgr3stgr 47942. (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 47924 . . . 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 11255 . . . . . . . . . . . . 13 0 ∈ V
54a1i 11 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 0 ∈ V)
6 prssg 4819 . . . . . . . . . . . 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 47921 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
141, 13ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘(StarGr‘𝑁)) = (0...𝑁)
1510, 12, 143eqtri 2769 . . . . . . . . . . . . . . . . . . 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 1095 . . . . . . . . . . . . 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 6992 . . . . . . . . 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 47816 . . . . . . . . . . . . . . . . 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 2852 . . . . . . . . . . . . . . 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 7298 . . . . . . . . . . . . 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 2852 . . . . . . . . . . . . . . 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 13663 . . . . . . . . . . . . . . . . . 18 (1...𝑁) ⊆ (0...𝑁)
4948sseli 3979 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1...𝑁) → 𝑦 ∈ (0...𝑁))
5049, 15eleqtrrdi 2852 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1...𝑁) → 𝑦𝑊)
5150adantl 481 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → 𝑦𝑊)
5247, 51ffvelcdmd 7105 . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . 17 ((𝐹𝑦) ∈ 𝐶 ↔ (𝐹𝑦) ∈ (𝐺 ClNeighbVtx 𝑋))
55 usgrupgr 29202 . . . . . . . . . . . . . . . . . . . . . 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 47818 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
5932, 58eqsstri 4030 . . . . . . . . . . . . . . . . . . . . 21 𝐶𝑉
6059, 52sselid 3981 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) ∧ 𝑦 ∈ (1...𝑁)) → (𝐹𝑦) ∈ 𝑉)
61 df-3an 1089 . . . . . . . . . . . . . . . . . . . 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 47821 . . . . . . . . . . . . . . . . . 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 2749 . . . . . . . . . . . . . . . . . . . 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 13664 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
741, 73ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ (0...𝑁)
7574, 15eleqtrri 2840 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ 𝑊
7650, 75jctir 520 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (1...𝑁) → (𝑦𝑊 ∧ 0 ∈ 𝑊))
77 f1veqaeq 7277 . . . . . . . . . . . . . . . . . . . . . 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 13593 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
80 nnne0 12300 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
81 eqneqall 2951 . . . . . . . . . . . . . . . . . . . . . . . 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 4732 . . . . . . . . . . . . . . . . . . . 20 {(𝐹𝑦), 𝑋} = {𝑋, (𝐹𝑦)}
9089eleq1i 2832 . . . . . . . . . . . . . . . . . . 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 860 . . . . . . . . . . . . . . . 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 4821 . . . . . . . . . . . . . . 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 4733 . . . . . . . . . . . . . . 15 ((𝐹‘0) = 𝑋 → {(𝐹‘0), (𝐹𝑦)} = {𝑋, (𝐹𝑦)})
101100eleq1d 2826 . . . . . . . . . . . . . 14 ((𝐹‘0) = 𝑋 → ({(𝐹‘0), (𝐹𝑦)} ∈ 𝐸 ↔ {𝑋, (𝐹𝑦)} ∈ 𝐸))
102100sseq1d 4015 . . . . . . . . . . . . . 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 29203 . . . . . . . . . . 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 2737 . . . . . . . . . . 11 (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
112 isubgr3stgr.i . . . . . . . . . . 11 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
11329, 64, 111, 112isubgredg 47852 . . . . . . . . . 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 2841 . . . . . . 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 4009 . . . . . . 7 (𝐽 = {0, 𝑦} → (𝐽 ⊆ (0...𝑁) ↔ {0, 𝑦} ⊆ (0...𝑁)))
119 imaeq2 6074 . . . . . . . 8 (𝐽 = {0, 𝑦} → (𝐹𝐽) = (𝐹 “ {0, 𝑦}))
120119eleq1d 2826 . . . . . . 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 3155 . . . 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 1118 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 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  wss 3951  {cpr 4628  cmpt 5225  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  cn 12266  0cn0 12526  ...cfz 13547  Vtxcvtx 29013  Edgcedg 29064  UHGraphcuhgr 29073  UPGraphcupgr 29097  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-uspgr 29167  df-usgr 29168  df-nbgr 29350  df-clnbgr 47806  df-isubgr 47847  df-stgr 47919
This theorem is referenced by:  isubgr3stgrlem8  47940
  Copyright terms: Public domain W3C validator