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Theorem 2pthfrgra 26304
Description: Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
Assertion
Ref Expression
2pthfrgra (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2))
Distinct variable groups:   𝑉,𝑎,𝑏,𝑓,𝑝   𝐸,𝑎,𝑏,𝑓,𝑝

Proof of Theorem 2pthfrgra
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 2pthfrgrarn2 26303 . 2 (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑚𝑉 (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)))
2 frisusgra 26285 . . . . . . . . . 10 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
3 usgrav 25633 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
42, 3syl 17 . . . . . . . . 9 (𝑉 FriendGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
54ad2antrr 757 . . . . . . . 8 (((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
65ad2antrr 757 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
7 simpr 475 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸𝑎𝑉) → 𝑎𝑉)
87ad2antrr 757 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → 𝑎𝑉)
9 simpr 475 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → 𝑚𝑉)
10 eldifsn 4259 . . . . . . . . . . 11 (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑏𝑉𝑏𝑎))
11 simpl 471 . . . . . . . . . . 11 ((𝑏𝑉𝑏𝑎) → 𝑏𝑉)
1210, 11sylbi 205 . . . . . . . . . 10 (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏𝑉)
1312ad2antlr 758 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → 𝑏𝑉)
148, 9, 133jca 1234 . . . . . . . 8 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → (𝑎𝑉𝑚𝑉𝑏𝑉))
1514adantr 479 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝑎𝑉𝑚𝑉𝑏𝑉))
16 simpll 785 . . . . . . . . . . . . . . 15 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → 𝑎𝑚)
17 necom 2834 . . . . . . . . . . . . . . . . 17 (𝑏𝑎𝑎𝑏)
1817biimpi 204 . . . . . . . . . . . . . . . 16 (𝑏𝑎𝑎𝑏)
1918adantl 480 . . . . . . . . . . . . . . 15 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → 𝑎𝑏)
20 simplr 787 . . . . . . . . . . . . . . 15 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → 𝑚𝑏)
2116, 19, 203jca 1234 . . . . . . . . . . . . . 14 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → (𝑎𝑚𝑎𝑏𝑚𝑏))
2221ex 448 . . . . . . . . . . . . 13 ((𝑎𝑚𝑚𝑏) → (𝑏𝑎 → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2322adantl 480 . . . . . . . . . . . 12 ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑏𝑎 → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2423com12 32 . . . . . . . . . . 11 (𝑏𝑎 → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2524adantl 480 . . . . . . . . . 10 ((𝑏𝑉𝑏𝑎) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2610, 25sylbi 205 . . . . . . . . 9 (𝑏 ∈ (𝑉 ∖ {𝑎}) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2726ad2antlr 758 . . . . . . . 8 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2827imp 443 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝑎𝑚𝑎𝑏𝑚𝑏))
29 usgraf1o 25653 . . . . . . . . . 10 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
30 fveq2 6088 . . . . . . . . . . . . . . . . 17 ((𝐸‘{𝑎, 𝑚}) = (𝐸‘{𝑚, 𝑏}) → (𝐸‘(𝐸‘{𝑎, 𝑚})) = (𝐸‘(𝐸‘{𝑚, 𝑏})))
31 simpl 471 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
32 simpll 785 . . . . . . . . . . . . . . . . . . . 20 ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → {𝑎, 𝑚} ∈ ran 𝐸)
33 f1ocnvfv2 6411 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑎, 𝑚} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚})
3431, 32, 33syl2an 492 . . . . . . . . . . . . . . . . . . 19 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚})
35 simplr 787 . . . . . . . . . . . . . . . . . . . 20 ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → {𝑚, 𝑏} ∈ ran 𝐸)
36 f1ocnvfv2 6411 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏})
3731, 35, 36syl2an 492 . . . . . . . . . . . . . . . . . . 19 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏})
3834, 37eqeq12d 2624 . . . . . . . . . . . . . . . . . 18 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → ((𝐸‘(𝐸‘{𝑎, 𝑚})) = (𝐸‘(𝐸‘{𝑚, 𝑏})) ↔ {𝑎, 𝑚} = {𝑚, 𝑏}))
39 prcom 4210 . . . . . . . . . . . . . . . . . . . . 21 {𝑚, 𝑏} = {𝑏, 𝑚}
4039eqeq2i 2621 . . . . . . . . . . . . . . . . . . . 20 ({𝑎, 𝑚} = {𝑚, 𝑏} ↔ {𝑎, 𝑚} = {𝑏, 𝑚})
41 vex 3175 . . . . . . . . . . . . . . . . . . . . 21 𝑎 ∈ V
42 vex 3175 . . . . . . . . . . . . . . . . . . . . 21 𝑏 ∈ V
4341, 42preqr1 4314 . . . . . . . . . . . . . . . . . . . 20 ({𝑎, 𝑚} = {𝑏, 𝑚} → 𝑎 = 𝑏)
4440, 43sylbi 205 . . . . . . . . . . . . . . . . . . 19 ({𝑎, 𝑚} = {𝑚, 𝑏} → 𝑎 = 𝑏)
45 nesym 2837 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏𝑎 ↔ ¬ 𝑎 = 𝑏)
46 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑎 = 𝑏 → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
4745, 46sylbi 205 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏𝑎 → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
4847adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑉𝑏𝑎) → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
4910, 48sylbi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5049adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5150ad2antlr 758 . . . . . . . . . . . . . . . . . . 19 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5244, 51syl5 33 . . . . . . . . . . . . . . . . . 18 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → ({𝑎, 𝑚} = {𝑚, 𝑏} → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5338, 52sylbid 228 . . . . . . . . . . . . . . . . 17 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → ((𝐸‘(𝐸‘{𝑎, 𝑚})) = (𝐸‘(𝐸‘{𝑚, 𝑏})) → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5430, 53syl5com 31 . . . . . . . . . . . . . . . 16 ((𝐸‘{𝑎, 𝑚}) = (𝐸‘{𝑚, 𝑏}) → (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
55 df-ne 2781 . . . . . . . . . . . . . . . . . 18 ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ↔ ¬ (𝐸‘{𝑎, 𝑚}) = (𝐸‘{𝑚, 𝑏}))
5655biimpri 216 . . . . . . . . . . . . . . . . 17 (¬ (𝐸‘{𝑎, 𝑚}) = (𝐸‘{𝑚, 𝑏}) → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}))
5756a1d 25 . . . . . . . . . . . . . . . 16 (¬ (𝐸‘{𝑎, 𝑚}) = (𝐸‘{𝑚, 𝑏}) → (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5854, 57pm2.61i 174 . . . . . . . . . . . . . . 15 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}))
5958, 34, 373jca 1234 . . . . . . . . . . . . . 14 (((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏}))
6059ex 448 . . . . . . . . . . . . 13 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏})))
6160expcom 449 . . . . . . . . . . . 12 (((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏}))))
6261exp31 627 . . . . . . . . . . 11 (𝑚𝑉 → (𝑎𝑉 → (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏}))))))
6362com14 93 . . . . . . . . . 10 (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (𝑎𝑉 → (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝑚𝑉 → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏}))))))
642, 29, 633syl 18 . . . . . . . . 9 (𝑉 FriendGrph 𝐸 → (𝑎𝑉 → (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝑚𝑉 → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏}))))))
6564imp 443 . . . . . . . 8 ((𝑉 FriendGrph 𝐸𝑎𝑉) → (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝑚𝑉 → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏})))))
6665imp41 616 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏}))
67 2pthon3v 25900 . . . . . . 7 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑎𝑉𝑚𝑉𝑏𝑉) ∧ (𝑎𝑚𝑎𝑏𝑚𝑏)) ∧ ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ∧ (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚} ∧ (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏})) → ∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2))
686, 15, 28, 66, 67syl31anc 1320 . . . . . 6 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → ∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2))
6968ex 448 . . . . 5 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2)))
7069rexlimdva 3012 . . . 4 (((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (∃𝑚𝑉 (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2)))
7170ralimdva 2944 . . 3 ((𝑉 FriendGrph 𝐸𝑎𝑉) → (∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑚𝑉 (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2)))
7271ralimdva 2944 . 2 (𝑉 FriendGrph 𝐸 → (∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑚𝑉 (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2)))
731, 72mpd 15 1 (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  {csn 4124  {cpr 4126   class class class wbr 4577  ccnv 5027  dom cdm 5028  ran crn 5029  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  2c2 10917  #chash 12934   USGrph cusg 25625   PathOn cpthon 25798   FriendGrph cfrgra 26281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-usgra 25628  df-wlk 25802  df-trail 25803  df-pth 25804  df-wlkon 25808  df-pthon 25810  df-frgra 26282
This theorem is referenced by:  frconngra  26314
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