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Theorem isrgrac 26461
 Description: The property of being a k-regular graph represented by a class. (Contributed by AV, 3-Jan-2020.)
Assertion
Ref Expression
isrgrac ((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))
Distinct variable groups:   𝐺,𝑝   𝐾,𝑝
Allowed substitution hints:   𝑋(𝑝)   𝑌(𝑝)   𝑍(𝑝)

Proof of Theorem isrgrac
StepHypRef Expression
1 1st2nd2 7096 . 2 (𝐺 ∈ (𝑋 × 𝑌) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
2 fvex 6113 . . . . 5 (1st𝐺) ∈ V
32a1i 11 . . . 4 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (1st𝐺) ∈ V)
4 fvex 6113 . . . . 5 (2nd𝐺) ∈ V
54a1i 11 . . . 4 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (2nd𝐺) ∈ V)
6 simpr 476 . . . 4 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → 𝐾𝑍)
7 isrgra 26453 . . . 4 (((1st𝐺) ∈ V ∧ (2nd𝐺) ∈ V ∧ 𝐾𝑍) → (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(((1st𝐺) VDeg (2nd𝐺))‘𝑝) = 𝐾)))
83, 5, 6, 7syl3anc 1318 . . 3 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(((1st𝐺) VDeg (2nd𝐺))‘𝑝) = 𝐾)))
9 breq1 4586 . . . . 5 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (𝐺 RegGrph 𝐾 ↔ ⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾))
10 fveq2 6103 . . . . . . . . . 10 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → ( VDeg ‘𝐺) = ( VDeg ‘⟨(1st𝐺), (2nd𝐺)⟩))
11 df-ov 6552 . . . . . . . . . 10 ((1st𝐺) VDeg (2nd𝐺)) = ( VDeg ‘⟨(1st𝐺), (2nd𝐺)⟩)
1210, 11syl6eqr 2662 . . . . . . . . 9 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → ( VDeg ‘𝐺) = ((1st𝐺) VDeg (2nd𝐺)))
1312fveq1d 6105 . . . . . . . 8 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (( VDeg ‘𝐺)‘𝑝) = (((1st𝐺) VDeg (2nd𝐺))‘𝑝))
1413eqeq1d 2612 . . . . . . 7 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → ((( VDeg ‘𝐺)‘𝑝) = 𝐾 ↔ (((1st𝐺) VDeg (2nd𝐺))‘𝑝) = 𝐾))
1514ralbidv 2969 . . . . . 6 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → (∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾 ↔ ∀𝑝 ∈ (1st𝐺)(((1st𝐺) VDeg (2nd𝐺))‘𝑝) = 𝐾))
1615anbi2d 736 . . . . 5 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → ((𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾) ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(((1st𝐺) VDeg (2nd𝐺))‘𝑝) = 𝐾)))
179, 16bibi12d 334 . . . 4 (𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ → ((𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)) ↔ (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(((1st𝐺) VDeg (2nd𝐺))‘𝑝) = 𝐾))))
1817adantr 480 . . 3 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → ((𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)) ↔ (⟨(1st𝐺), (2nd𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(((1st𝐺) VDeg (2nd𝐺))‘𝑝) = 𝐾))))
198, 18mpbird 246 . 2 ((𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩ ∧ 𝐾𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))
201, 19sylan 487 1 ((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ℕ0cn0 11169   VDeg cvdg 26420   RegGrph crgra 26449 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 This theorem is referenced by:  isrusgrac  26462
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