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Theorem vdgrfiun 26429

Description: The degree of a vertex in the union of two graphs (of finite size) on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)

**Hypotheses**
Ref	Expression
vdgrfiun.e	⊢ (𝜑 → 𝐸 Fn 𝐴)
vdgrfiun.f	⊢ (𝜑 → 𝐹 Fn 𝐵)
vdgrfiun.a	⊢ (𝜑 → 𝐴 ∈ Fin)
vdgrfiun.b	⊢ (𝜑 → 𝐵 ∈ Fin)
vdgrfiun.i	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
vdgrfiun.ge	⊢ (𝜑 → 𝑉 UMGrph 𝐸)
vdgrfiun.gf	⊢ (𝜑 → 𝑉 UMGrph 𝐹)
vdgrfiun.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)

**Assertion**
Ref	Expression
vdgrfiun	⊢ (𝜑 → ((𝑉 VDeg (𝐸 ∪ 𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)))

Proof of Theorem vdgrfiun Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3715	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2	1	anbi1i 727	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)))
3		andir 908	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)) ∨ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))))
4	2, 3	bitri 263	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)) ∨ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))))
5	4	abbii 2726	. . . . . . . 8 ⊢ {𝑥 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)) ∨ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)))}
6		df-rab 2905	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))}
7		unab 3853	. . . . . . . 8 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)) ∨ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)))}
8	5, 6, 7	3eqtr4i 2642	. . . . . . 7 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))})
9		df-rab 2905	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))}
10		vdgrfiun.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 Fn 𝐴)
11	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐸 Fn 𝐴)
12		vdgrfiun.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn 𝐵)
13	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐵)
14		vdgrfiun.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
15	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∩ 𝐵) = ∅)
16		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
17		fvun1 6179	. . . . . . . . . . . 12 ⊢ ((𝐸 Fn 𝐴 ∧ 𝐹 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴)) → ((𝐸 ∪ 𝐹)‘𝑥) = (𝐸‘𝑥))
18	11, 13, 15, 16, 17	syl112anc 1322	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐸 ∪ 𝐹)‘𝑥) = (𝐸‘𝑥))
19	18	eleq2d 2673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐸‘𝑥)))
20	19	rabbidva 3163	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)})
21	9, 20	syl5eqr 2658	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))} = {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)})
22		df-rab 2905	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))}
23	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 Fn 𝐴)
24	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 Fn 𝐵)
25	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) = ∅)
26		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
27		fvun2 6180	. . . . . . . . . . . 12 ⊢ ((𝐸 Fn 𝐴 ∧ 𝐹 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐵)) → ((𝐸 ∪ 𝐹)‘𝑥) = (𝐹‘𝑥))
28	23, 24, 25, 26, 27	syl112anc 1322	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐸 ∪ 𝐹)‘𝑥) = (𝐹‘𝑥))
29	28	eleq2d 2673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐹‘𝑥)))
30	29	rabbidva 3163	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)} = {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})
31	22, 30	syl5eqr 2658	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))} = {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})
32	21, 31	uneq12d 3730	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥))}) = ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}))
33	8, 32	syl5eq 2656	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)} = ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}))
34	33	fveq2d 6107	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)}) = (#‘({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})))
35		vdgrfiun.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin)
36		ssrab2 3650	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ⊆ 𝐴
37		ssfi 8065	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ⊆ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ Fin)
38	35, 36, 37	sylancl 693	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ Fin)
39		vdgrfiun.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
40		ssrab2 3650	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)} ⊆ 𝐵
41		ssfi 8065	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)} ⊆ 𝐵) → {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)} ∈ Fin)
42	39, 40, 41	sylancl 693	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)} ∈ Fin)
43		ss2in 3802	. . . . . . . . 9 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)} ⊆ 𝐵) → ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∩ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) ⊆ (𝐴 ∩ 𝐵))
44	36, 40, 43	mp2an 704	. . . . . . . 8 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∩ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) ⊆ (𝐴 ∩ 𝐵)
45	44, 14	syl5sseq 3616	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∩ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) ⊆ ∅)
46		ss0 3926	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∩ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) ⊆ ∅ → ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∩ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) = ∅)
47	45, 46	syl 17	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∩ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) = ∅)
48		hashun 13032	. . . . . 6 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ Fin ∧ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)} ∈ Fin ∧ ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∩ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) = ∅) → (#‘({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})))
49	38, 42, 47, 48	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (#‘({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})))
50	34, 49	eqtrd 2644	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)}) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})))
51	1	anbi1i 727	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}))
52		andir 908	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}) ∨ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})))
53	51, 52	bitri 263	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}) ∨ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})))
54	53	abbii 2726	. . . . . . . 8 ⊢ {𝑥 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}) ∨ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}))}
55		df-rab 2905	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})}
56		unab 3853	. . . . . . . 8 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}) ∨ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}))}
57	54, 55, 56	3eqtr4i 2642	. . . . . . 7 ⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})})
58		df-rab 2905	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})}
59	18	eqeq1d 2612	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝐸 ∪ 𝐹)‘𝑥) = {𝑈} ↔ (𝐸‘𝑥) = {𝑈}))
60	59	rabbidva 3163	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}} = {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})
61	58, 60	syl5eqr 2658	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})} = {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})
62		df-rab 2905	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐵 ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})}
63	28	eqeq1d 2612	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐸 ∪ 𝐹)‘𝑥) = {𝑈} ↔ (𝐹‘𝑥) = {𝑈}))
64	63	rabbidva 3163	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}} = {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})
65	62, 64	syl5eqr 2658	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})} = {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})
66	61, 65	uneq12d 3730	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈})}) = ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∪ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}))
67	57, 66	syl5eq 2656	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}} = ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∪ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}))
68	67	fveq2d 6107	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}}) = (#‘({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∪ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})))
69		ssrab2 3650	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ⊆ 𝐴
70		ssfi 8065	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ⊆ 𝐴) → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∈ Fin)
71	35, 69, 70	sylancl 693	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∈ Fin)
72		ssrab2 3650	. . . . . . 7 ⊢ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}} ⊆ 𝐵
73		ssfi 8065	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}} ⊆ 𝐵) → {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}} ∈ Fin)
74	39, 72, 73	sylancl 693	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}} ∈ Fin)
75		ss2in 3802	. . . . . . . . 9 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}} ⊆ 𝐵) → ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∩ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) ⊆ (𝐴 ∩ 𝐵))
76	69, 72, 75	mp2an 704	. . . . . . . 8 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∩ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) ⊆ (𝐴 ∩ 𝐵)
77	76, 14	syl5sseq 3616	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∩ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) ⊆ ∅)
78		ss0 3926	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∩ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) ⊆ ∅ → ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∩ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) = ∅)
79	77, 78	syl 17	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∩ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) = ∅)
80		hashun 13032	. . . . . 6 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∈ Fin ∧ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}} ∈ Fin ∧ ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∩ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) = ∅) → (#‘({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∪ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})) = ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})))
81	71, 74, 79, 80	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (#‘({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∪ {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})) = ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})))
82	68, 81	eqtrd 2644	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}}) = ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})))
83	50, 82	oveq12d 6567	. . 3 ⊢ (𝜑 → ((#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})) + ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}))))
84		hashcl 13009	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℕ₀)
85	38, 84	syl 17	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℕ₀)
86	85	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℂ)
87		hashcl 13009	. . . . . 6 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)} ∈ Fin → (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) ∈ ℕ₀)
88	42, 87	syl 17	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) ∈ ℕ₀)
89	88	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) ∈ ℂ)
90		hashcl 13009	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℕ₀)
91	71, 90	syl 17	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℕ₀)
92	91	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℂ)
93		hashcl 13009	. . . . . 6 ⊢ ({𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}} ∈ Fin → (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) ∈ ℕ₀)
94	74, 93	syl 17	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) ∈ ℕ₀)
95	94	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}) ∈ ℂ)
96	86, 89, 92, 95	add4d 10143	. . 3 ⊢ (𝜑 → (((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)})) + ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}))) = (((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})) + ((#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}))))
97	83, 96	eqtrd 2644	. 2 ⊢ (𝜑 → ((#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})) + ((#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}))))
98		relumgra 25843	. . . 4 ⊢ Rel UMGrph
99		vdgrfiun.ge	. . . 4 ⊢ (𝜑 → 𝑉 UMGrph 𝐸)
100		brrelex 5080	. . . 4 ⊢ ((Rel UMGrph ∧ 𝑉 UMGrph 𝐸) → 𝑉 ∈ V)
101	98, 99, 100	sylancr 694	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
102		fnun 5911	. . . 4 ⊢ (((𝐸 Fn 𝐴 ∧ 𝐹 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐸 ∪ 𝐹) Fn (𝐴 ∪ 𝐵))
103	10, 12, 14, 102	syl21anc 1317	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) Fn (𝐴 ∪ 𝐵))
104		unfi 8112	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
105	35, 39, 104	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ Fin)
106		vdgrfiun.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
107		vdgrfival 26424	. . 3 ⊢ (((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) Fn (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg (𝐸 ∪ 𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}})))
108	101, 103, 105, 106, 107	syl31anc 1321	. 2 ⊢ (𝜑 → ((𝑉 VDeg (𝐸 ∪ 𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝑈 ∈ ((𝐸 ∪ 𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴 ∪ 𝐵) ∣ ((𝐸 ∪ 𝐹)‘𝑥) = {𝑈}})))
109		vdgrfival 26424	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})))
110	101, 10, 35, 106, 109	syl31anc 1321	. . 3 ⊢ (𝜑 → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})))
111		vdgrfival 26424	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐹 Fn 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})))
112	101, 12, 39, 106, 111	syl31anc 1321	. . 3 ⊢ (𝜑 → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}})))
113	110, 112	oveq12d 6567	. 2 ⊢ (𝜑 → (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)) = (((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})) + ((#‘{𝑥 ∈ 𝐵 ∣ 𝑈 ∈ (𝐹‘𝑥)}) + (#‘{𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) = {𝑈}}))))
114	97, 108, 113	3eqtr4d 2654	1 ⊢ (𝜑 → ((𝑉 VDeg (𝐸 ∪ 𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 {crab 2900 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 Rel wrel 5043 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 + caddc 9818 ℕ₀cn0 11169 #chash 12979 UMGrph cumg 25841 VDeg cvdg 26420

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-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-n0 11170 df-z 11255 df-uz 11564 df-xadd 11823 df-hash 12980 df-umgra 25842 df-vdgr 26421

This theorem is referenced by: eupath2lem3 26506 vdegp1ai 26511 vdegp1bi 26512

W3C validator