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Theorem 2pthfrgra 26538
 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 26537 . 2 (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑚𝑉 (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)))
2 frisusgra 26519 . . . . . . . . . 10 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
3 usgrav 25867 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
42, 3syl 17 . . . . . . . . 9 (𝑉 FriendGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
54ad2antrr 758 . . . . . . . 8 (((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
65ad2antrr 758 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
7 simpr 476 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸𝑎𝑉) → 𝑎𝑉)
87ad2antrr 758 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → 𝑎𝑉)
9 simpr 476 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → 𝑚𝑉)
10 eldifsn 4260 . . . . . . . . . . 11 (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑏𝑉𝑏𝑎))
11 simpl 472 . . . . . . . . . . 11 ((𝑏𝑉𝑏𝑎) → 𝑏𝑉)
1210, 11sylbi 206 . . . . . . . . . 10 (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏𝑉)
1312ad2antlr 759 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → 𝑏𝑉)
148, 9, 133jca 1235 . . . . . . . 8 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → (𝑎𝑉𝑚𝑉𝑏𝑉))
1514adantr 480 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝑎𝑉𝑚𝑉𝑏𝑉))
16 simpll 786 . . . . . . . . . . . . . . 15 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → 𝑎𝑚)
17 necom 2835 . . . . . . . . . . . . . . . . 17 (𝑏𝑎𝑎𝑏)
1817biimpi 205 . . . . . . . . . . . . . . . 16 (𝑏𝑎𝑎𝑏)
1918adantl 481 . . . . . . . . . . . . . . 15 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → 𝑎𝑏)
20 simplr 788 . . . . . . . . . . . . . . 15 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → 𝑚𝑏)
2116, 19, 203jca 1235 . . . . . . . . . . . . . 14 (((𝑎𝑚𝑚𝑏) ∧ 𝑏𝑎) → (𝑎𝑚𝑎𝑏𝑚𝑏))
2221ex 449 . . . . . . . . . . . . 13 ((𝑎𝑚𝑚𝑏) → (𝑏𝑎 → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2322adantl 481 . . . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝑏𝑉𝑏𝑎) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2610, 25sylbi 206 . . . . . . . . 9 (𝑏 ∈ (𝑉 ∖ {𝑎}) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2726ad2antlr 759 . . . . . . . 8 ((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) → ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → (𝑎𝑚𝑎𝑏𝑚𝑏)))
2827imp 444 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ∧ 𝑚𝑉) ∧ (({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏))) → (𝑎𝑚𝑎𝑏𝑚𝑏))
29 usgraf1o 25887 . . . . . . . . . 10 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
30 fveq2 6103 . . . . . . . . . . . . . . . . 17 ((𝐸‘{𝑎, 𝑚}) = (𝐸‘{𝑚, 𝑏}) → (𝐸‘(𝐸‘{𝑎, 𝑚})) = (𝐸‘(𝐸‘{𝑚, 𝑏})))
31 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎}))) → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
32 simpll 786 . . . . . . . . . . . . . . . . . . . 20 ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → {𝑎, 𝑚} ∈ ran 𝐸)
33 f1ocnvfv2 6433 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑎, 𝑚} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑎, 𝑚})) = {𝑎, 𝑚})
3431, 32, 33syl2an 493 . . . . . . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . . . . . 20 ((({𝑎, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) ∧ (𝑎𝑚𝑚𝑏)) → {𝑚, 𝑏} ∈ ran 𝐸)
36 f1ocnvfv2 6433 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑚, 𝑏} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑚, 𝑏})) = {𝑚, 𝑏})
3731, 35, 36syl2an 493 . . . . . . . . . . . . . . . . . . 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 2625 . . . . . . . . . . . . . . . . . 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 4211 . . . . . . . . . . . . . . . . . . . . 21 {𝑚, 𝑏} = {𝑏, 𝑚}
4039eqeq2i 2622 . . . . . . . . . . . . . . . . . . . 20 ({𝑎, 𝑚} = {𝑚, 𝑏} ↔ {𝑎, 𝑚} = {𝑏, 𝑚})
41 vex 3176 . . . . . . . . . . . . . . . . . . . . 21 𝑎 ∈ V
42 vex 3176 . . . . . . . . . . . . . . . . . . . . 21 𝑏 ∈ V
4341, 42preqr1 4319 . . . . . . . . . . . . . . . . . . . 20 ({𝑎, 𝑚} = {𝑏, 𝑚} → 𝑎 = 𝑏)
4440, 43sylbi 206 . . . . . . . . . . . . . . . . . . 19 ({𝑎, 𝑚} = {𝑚, 𝑏} → 𝑎 = 𝑏)
45 nesym 2838 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏𝑎 ↔ ¬ 𝑎 = 𝑏)
46 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑎 = 𝑏 → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
4745, 46sylbi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏𝑎 → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
4847adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏𝑉𝑏𝑎) → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
4910, 48sylbi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5049adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑚𝑉𝑎𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎 = 𝑏 → (𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏})))
5150ad2antlr 759 . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . . . . . . 18 ((𝐸‘{𝑎, 𝑚}) ≠ (𝐸‘{𝑚, 𝑏}) ↔ ¬ (𝐸‘{𝑎, 𝑚}) = (𝐸‘{𝑚, 𝑏}))
5655biimpri 217 . . . . . . . . . . . . . . . . 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 175 . . . . . . . . . . . . . . 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 1235 . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 628 . . . . . . . . . . 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 94 . . . . . . . . . 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 444 . . . . . . . 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 617 . . . . . . 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 26134 . . . . . . 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 1321 . . . . . 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 449 . . . . 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 3013 . . . 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 2945 . . 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 2945 . 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 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  2c2 10947  #chash 12979   USGrph cusg 25859   PathOn cpthon 26032   FriendGrph cfrgra 26515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-wlkon 26042  df-pthon 26044  df-frgra 26516 This theorem is referenced by:  frconngra  26548
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