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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) ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))

Distinct variable groups: 𝐺,𝑝 𝐾,𝑝 Allowed substitution hints: 𝑋(𝑝) 𝑌(𝑝) 𝑍(𝑝)

**Proof of Theorem isrgrac**
Step	Hyp	Ref	Expression
1		1st2nd2 7096	. 2 ⊢ (𝐺 ∈ (𝑋 × 𝑌) → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
2		fvex 6113	. . . . 5 ⊢ (1^st ‘𝐺) ∈ V
3	2	a1i 11	. . . 4 ⊢ ((𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ ∧ 𝐾 ∈ 𝑍) → (1^st ‘𝐺) ∈ V)
4		fvex 6113	. . . . 5 ⊢ (2^nd ‘𝐺) ∈ V
5	4	a1i 11	. . . 4 ⊢ ((𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ ∧ 𝐾 ∈ 𝑍) → (2^nd ‘𝐺) ∈ V)
6		simpr 476	. . . 4 ⊢ ((𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ ∧ 𝐾 ∈ 𝑍) → 𝐾 ∈ 𝑍)
7		isrgra 26453	. . . 4 ⊢ (((1^st ‘𝐺) ∈ V ∧ (2^nd ‘𝐺) ∈ V ∧ 𝐾 ∈ 𝑍) → (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝) = 𝐾)))
8	3, 5, 6, 7	syl3anc 1318	. . 3 ⊢ ((𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ ∧ 𝐾 ∈ 𝑍) → (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝) = 𝐾)))
9		breq1 4586	. . . . 5 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → (𝐺 RegGrph 𝐾 ↔ ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ RegGrph 𝐾))
10		fveq2 6103	. . . . . . . . . 10 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → ( VDeg ‘𝐺) = ( VDeg ‘⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
11		df-ov 6552	. . . . . . . . . 10 ⊢ ((1^st ‘𝐺) VDeg (2^nd ‘𝐺)) = ( VDeg ‘⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
12	10, 11	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → ( VDeg ‘𝐺) = ((1^st ‘𝐺) VDeg (2^nd ‘𝐺)))
13	12	fveq1d 6105	. . . . . . . 8 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → (( VDeg ‘𝐺)‘𝑝) = (((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝))
14	13	eqeq1d 2612	. . . . . . 7 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → ((( VDeg ‘𝐺)‘𝑝) = 𝐾 ↔ (((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝) = 𝐾))
15	14	ralbidv 2969	. . . . . 6 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → (∀𝑝 ∈ (1^st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾 ↔ ∀𝑝 ∈ (1^st ‘𝐺)(((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝) = 𝐾))
16	15	anbi2d 736	. . . . 5 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → ((𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾) ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝) = 𝐾)))
17	9, 16	bibi12d 334	. . . 4 ⊢ (𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ → ((𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)) ↔ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝) = 𝐾))))
18	17	adantr 480	. . 3 ⊢ ((𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ ∧ 𝐾 ∈ 𝑍) → ((𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)) ↔ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(((1^st ‘𝐺) VDeg (2^nd ‘𝐺))‘𝑝) = 𝐾))))
19	8, 18	mpbird 246	. 2 ⊢ ((𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩ ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))
20	1, 19	sylan 487	1 ⊢ ((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ₀ ∧ ∀𝑝 ∈ (1^st ‘𝐺)(( 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 1^st c1st 7057 2^nd 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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