Theorem frgrusgrfrcond 27123

Description: A friendship graph is a simple graph which fulfils the friendship condition. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
isfrgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isfrgr.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frgrusgrfrcond	⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

Distinct variable groups:   𝑘,𝑙,𝑥,𝐺   𝑘,𝑉,𝑙,𝑥

Allowed substitution hints:   𝐸(𝑥,𝑘,𝑙)

Proof of Theorem frgrusgrfrcond

Step	Hyp	Ref	Expression
1		isfrgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		isfrgr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	isfrgr 27122	. . . 4 ⊢ (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
4		simpl 473	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸) → 𝐺 ∈ USGraph )
5	3, 4	syl6bi 243	. . 3 ⊢ (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph ))
6	5	pm2.43i 52	. 2 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
7	1, 2	isfrgr 27122	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
8	6, 4, 7	pm5.21nii 368	1 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃!wreu 2914   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  {cpr 4179  ‘cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-frgr 27121

This theorem is referenced by:  frgrusgr  27124  frgr0v  27125  frgr0  27128  frcond1  27130  frgr1v  27135  nfrgr2v  27136  frgr3v  27139  2pthfrgrrn  27146  n4cyclfrgr  27155