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Theorem 3vfriswmgra 26532
 Description: Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
3vfriswmgra (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
Distinct variable groups:   𝐴,,𝑣,𝑤   𝐵,,𝑣,𝑤   𝐶,,𝑣,𝑤   ,𝐸,𝑣,𝑤   ,𝑉,𝑣,𝑤   𝑣,𝑋,𝑤   𝑣,𝑌,𝑤
Allowed substitution hints:   𝑋()   𝑌()   𝑍(𝑤,𝑣,)

Proof of Theorem 3vfriswmgra
StepHypRef Expression
1 frisusgra 26519 . . . 4 ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → {𝐴, 𝐵, 𝐶} USGrph 𝐸)
2 frgra3v 26529 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
323adant3 1074 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
43imp 444 . . . . . . 7 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
5 prcom 4211 . . . . . . . . . . . . . . . . . 18 {𝐶, 𝐴} = {𝐴, 𝐶}
65eleq1i 2679 . . . . . . . . . . . . . . . . 17 ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸)
76biimpi 205 . . . . . . . . . . . . . . . 16 ({𝐶, 𝐴} ∈ ran 𝐸 → {𝐴, 𝐶} ∈ ran 𝐸)
873ad2ant3 1077 . . . . . . . . . . . . . . 15 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → {𝐴, 𝐶} ∈ ran 𝐸)
98adantl 481 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → {𝐴, 𝐶} ∈ ran 𝐸)
10 id 22 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑋𝐵𝑌) → (𝐴𝑋𝐵𝑌))
11103adant3 1074 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐴𝑋𝐵𝑌))
12113ad2ant1 1075 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐴𝑋𝐵𝑌))
1312adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (𝐴𝑋𝐵𝑌))
14 id 22 . . . . . . . . . . . . . . . . . . 19 (𝐴𝐵𝐴𝐵)
15143ad2ant1 1075 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐴𝐵)
16153ad2ant2 1076 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐴𝐵)
1716adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴𝐵)
18 simpr 476 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → {𝐴, 𝐵, 𝐶} USGrph 𝐸)
1913, 17, 183jca 1235 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸))
20 id 22 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} ∈ ran 𝐸 → {𝐴, 𝐵} ∈ ran 𝐸)
21203ad2ant1 1075 . . . . . . . . . . . . . . 15 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → {𝐴, 𝐵} ∈ ran 𝐸)
22 3vfriswmgralem 26531 . . . . . . . . . . . . . . . 16 (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
2322imp 444 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)
2419, 21, 23syl2an 493 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)
259, 24jca 553 . . . . . . . . . . . . 13 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
26 simpr2 1061 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → {𝐵, 𝐶} ∈ ran 𝐸)
27 pm3.22 464 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑋𝐵𝑌) → (𝐵𝑌𝐴𝑋))
28273adant3 1074 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐵𝑌𝐴𝑋))
29283ad2ant1 1075 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐵𝑌𝐴𝑋))
3029adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (𝐵𝑌𝐴𝑋))
31 necom 2835 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐵𝐴)
3231biimpi 205 . . . . . . . . . . . . . . . . . . 19 (𝐴𝐵𝐵𝐴)
33323ad2ant1 1075 . . . . . . . . . . . . . . . . . 18 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐵𝐴)
34333ad2ant2 1076 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐵𝐴)
3534adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵𝐴)
36 tpcoma 4229 . . . . . . . . . . . . . . . . . . 19 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
3736breq1i 4590 . . . . . . . . . . . . . . . . . 18 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ↔ {𝐵, 𝐴, 𝐶} USGrph 𝐸)
3837biimpi 205 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → {𝐵, 𝐴, 𝐶} USGrph 𝐸)
3938adantl 481 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → {𝐵, 𝐴, 𝐶} USGrph 𝐸)
4030, 35, 393jca 1235 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((𝐵𝑌𝐴𝑋) ∧ 𝐵𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸))
41 prcom 4211 . . . . . . . . . . . . . . . . . 18 {𝐴, 𝐵} = {𝐵, 𝐴}
4241eleq1i 2679 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸)
4342biimpi 205 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} ∈ ran 𝐸 → {𝐵, 𝐴} ∈ ran 𝐸)
44433ad2ant1 1075 . . . . . . . . . . . . . . 15 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → {𝐵, 𝐴} ∈ ran 𝐸)
45 3vfriswmgralem 26531 . . . . . . . . . . . . . . . . 17 (((𝐵𝑌𝐴𝑋) ∧ 𝐵𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) → ({𝐵, 𝐴} ∈ ran 𝐸 → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸))
4645imp 444 . . . . . . . . . . . . . . . 16 ((((𝐵𝑌𝐴𝑋) ∧ 𝐵𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸) → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸)
47 reueq1 3117 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐵} = {𝐵, 𝐴} → (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸))
4841, 47ax-mp 5 . . . . . . . . . . . . . . . 16 (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸)
4946, 48sylibr 223 . . . . . . . . . . . . . . 15 ((((𝐵𝑌𝐴𝑋) ∧ 𝐵𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)
5040, 44, 49syl2an 493 . . . . . . . . . . . . . 14 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)
5126, 50jca 553 . . . . . . . . . . . . 13 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))
5225, 51jca 553 . . . . . . . . . . . 12 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)))
53 preq1 4212 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → {𝑣, 𝐶} = {𝐴, 𝐶})
5453eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → ({𝑣, 𝐶} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸))
55 preq1 4212 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐴 → {𝑣, 𝑤} = {𝐴, 𝑤})
5655eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → ({𝑣, 𝑤} ∈ ran 𝐸 ↔ {𝐴, 𝑤} ∈ ran 𝐸))
5756reubidv 3103 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
5854, 57anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐴 → (({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)))
59 preq1 4212 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → {𝑣, 𝐶} = {𝐵, 𝐶})
6059eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → ({𝑣, 𝐶} ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸))
61 preq1 4212 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝐵 → {𝑣, 𝑤} = {𝐵, 𝑤})
6261eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐵 → ({𝑣, 𝑤} ∈ ran 𝐸 ↔ {𝐵, 𝑤} ∈ ran 𝐸))
6362reubidv 3103 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐵 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))
6460, 63anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)))
6558, 64ralprg 4181 . . . . . . . . . . . . . . . 16 ((𝐴𝑋𝐵𝑌) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
66653adant3 1074 . . . . . . . . . . . . . . 15 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
67663ad2ant1 1075 . . . . . . . . . . . . . 14 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
6867adantr 480 . . . . . . . . . . . . 13 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
6968adantr 480 . . . . . . . . . . . 12 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
7052, 69mpbird 246 . . . . . . . . . . 11 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸))
71 diftpsn3 4273 . . . . . . . . . . . . . . . 16 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
72713adant1 1072 . . . . . . . . . . . . . . 15 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
73 reueq1 3117 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵} → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸))
7472, 73syl 17 . . . . . . . . . . . . . . . 16 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸))
7574anbi2d 736 . . . . . . . . . . . . . . 15 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
7672, 75raleqbidv 3129 . . . . . . . . . . . . . 14 ((𝐴𝐵𝐴𝐶𝐵𝐶) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
77763ad2ant2 1076 . . . . . . . . . . . . 13 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
7877adantr 480 . . . . . . . . . . . 12 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
7978adantr 480 . . . . . . . . . . 11 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
8070, 79mpbird 246 . . . . . . . . . 10 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))
81803mix3d 1231 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸)))
82 sneq 4135 . . . . . . . . . . . . . . 15 ( = 𝐴 → {} = {𝐴})
8382difeq2d 3690 . . . . . . . . . . . . . 14 ( = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
84 preq2 4213 . . . . . . . . . . . . . . . 16 ( = 𝐴 → {𝑣, } = {𝑣, 𝐴})
8584eleq1d 2672 . . . . . . . . . . . . . . 15 ( = 𝐴 → ({𝑣, } ∈ ran 𝐸 ↔ {𝑣, 𝐴} ∈ ran 𝐸))
86 reueq1 3117 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸))
8783, 86syl 17 . . . . . . . . . . . . . . 15 ( = 𝐴 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸))
8885, 87anbi12d 743 . . . . . . . . . . . . . 14 ( = 𝐴 → (({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸)))
8983, 88raleqbidv 3129 . . . . . . . . . . . . 13 ( = 𝐴 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸)))
90 sneq 4135 . . . . . . . . . . . . . . 15 ( = 𝐵 → {} = {𝐵})
9190difeq2d 3690 . . . . . . . . . . . . . 14 ( = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
92 preq2 4213 . . . . . . . . . . . . . . . 16 ( = 𝐵 → {𝑣, } = {𝑣, 𝐵})
9392eleq1d 2672 . . . . . . . . . . . . . . 15 ( = 𝐵 → ({𝑣, } ∈ ran 𝐸 ↔ {𝑣, 𝐵} ∈ ran 𝐸))
94 reueq1 3117 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸))
9591, 94syl 17 . . . . . . . . . . . . . . 15 ( = 𝐵 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸))
9693, 95anbi12d 743 . . . . . . . . . . . . . 14 ( = 𝐵 → (({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸)))
9791, 96raleqbidv 3129 . . . . . . . . . . . . 13 ( = 𝐵 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸)))
98 sneq 4135 . . . . . . . . . . . . . . 15 ( = 𝐶 → {} = {𝐶})
9998difeq2d 3690 . . . . . . . . . . . . . 14 ( = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
100 preq2 4213 . . . . . . . . . . . . . . . 16 ( = 𝐶 → {𝑣, } = {𝑣, 𝐶})
101100eleq1d 2672 . . . . . . . . . . . . . . 15 ( = 𝐶 → ({𝑣, } ∈ ran 𝐸 ↔ {𝑣, 𝐶} ∈ ran 𝐸))
102 reueq1 3117 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵, 𝐶} ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))
10399, 102syl 17 . . . . . . . . . . . . . . 15 ( = 𝐶 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))
104101, 103anbi12d 743 . . . . . . . . . . . . . 14 ( = 𝐶 → (({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸)))
10599, 104raleqbidv 3129 . . . . . . . . . . . . 13 ( = 𝐶 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸)))
10689, 97, 105rextpg 4184 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
1071063ad2ant1 1075 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
108107adantr 480 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
109108adantr 480 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
11081, 109mpbird 246 . . . . . . . 8 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))
111110ex 449 . . . . . . 7 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
1124, 111sylbid 229 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
113112expcom 450 . . . . 5 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
114113com23 84 . . . 4 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
1151, 114mpcom 37 . . 3 ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
116115com12 32 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
117 breq1 4586 . . . 4 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 FriendGrph 𝐸 ↔ {𝐴, 𝐵, 𝐶} FriendGrph 𝐸))
118 difeq1 3683 . . . . . 6 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {}))
119 reueq1 3117 . . . . . . . 8 ((𝑉 ∖ {}) = ({𝐴, 𝐵, 𝐶} ∖ {}) → (∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))
120118, 119syl 17 . . . . . . 7 (𝑉 = {𝐴, 𝐵, 𝐶} → (∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))
121120anbi2d 736 . . . . . 6 (𝑉 = {𝐴, 𝐵, 𝐶} → (({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
122118, 121raleqbidv 3129 . . . . 5 (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
123122rexeqbi1dv 3124 . . . 4 (𝑉 = {𝐴, 𝐵, 𝐶} → (∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
124117, 123imbi12d 333 . . 3 (𝑉 = {𝐴, 𝐵, 𝐶} → ((𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
1251243ad2ant3 1077 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ((𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))))
126116, 125mpbird 246 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∖ cdif 3537  {csn 4125  {cpr 4127  {ctp 4129   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   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-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-hash 12980  df-usgra 25862  df-frgra 26516 This theorem is referenced by:  1to3vfriswmgra  26534
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