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Theorem isrusgusrgcl 26460
 Description: A graph represented by a class is a k-regular undirected simple graph iff it is an undirected simple graph and a k-regular graph. (Contributed by AV, 2-Jan-2020.)
Assertion
Ref Expression
isrusgusrgcl ((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾𝑍) → (𝐺 RegUSGrph 𝐾 ↔ (𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾)))

Proof of Theorem isrusgusrgcl
StepHypRef Expression
1 1st2nd2 7096 . 2 (𝐺 ∈ (𝑋 × 𝑌) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
2 breq1 4586 . . . 4 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 RegUSGrph 𝐾 ↔ ⟨(1st𝐺), (2nd𝐺)⟩ RegUSGrph 𝐾))
32adantr 480 . . 3 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (𝐺 RegUSGrph 𝐾 ↔ ⟨(1st𝐺), (2nd𝐺)⟩ RegUSGrph 𝐾))
4 fvex 6113 . . . . 5 (1st𝐺) ∈ V
54a1i 11 . . . 4 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (1st𝐺) ∈ V)
6 fvex 6113 . . . . 5 (2nd𝐺) ∈ V
76a1i 11 . . . 4 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (2nd𝐺) ∈ V)
8 simpr 476 . . . 4 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → 𝐾𝑍)
9 isrusgusrg 26459 . . . 4 (((1st𝐺) ∈ V ∧ (2nd𝐺) ∈ V ∧ 𝐾𝑍) → (⟨(1st𝐺), (2nd𝐺)⟩ RegUSGrph 𝐾 ↔ ((1st𝐺) USGrph (2nd𝐺) ∧ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾)))
105, 7, 8, 9syl3anc 1318 . . 3 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (⟨(1st𝐺), (2nd𝐺)⟩ RegUSGrph 𝐾 ↔ ((1st𝐺) USGrph (2nd𝐺) ∧ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾)))
11 eleq1 2676 . . . . . . 7 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ USGrph ))
12 df-br 4584 . . . . . . 7 ((1st𝐺) USGrph (2nd𝐺) ↔ ⟨(1st𝐺), (2nd𝐺)⟩ ∈ USGrph )
1311, 12syl6bbr 277 . . . . . 6 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 ∈ USGrph ↔ (1st𝐺) USGrph (2nd𝐺)))
14 breq1 4586 . . . . . 6 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 RegGrph 𝐾 ↔ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾))
1513, 14anbi12d 743 . . . . 5 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → ((𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾) ↔ ((1st𝐺) USGrph (2nd𝐺) ∧ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾)))
1615bicomd 212 . . . 4 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (((1st𝐺) USGrph (2nd𝐺) ∧ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾) ↔ (𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾)))
1716adantr 480 . . 3 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (((1st𝐺) USGrph (2nd𝐺) ∧ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾) ↔ (𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾)))
183, 10, 173bitrd 293 . 2 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (𝐺 RegUSGrph 𝐾 ↔ (𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾)))
191, 18sylan 487 1 ((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾𝑍) → (𝐺 RegUSGrph 𝐾 ↔ (𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058   USGrph cusg 25859   RegGrph crgra 26449   RegUSGrph crusgra 26450 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-1st 7059  df-2nd 7060  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  isrusgrac  26462
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