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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 (𝜑𝐸 Fn 𝐴)
vdgrfiun.f (𝜑𝐹 Fn 𝐵)
vdgrfiun.a (𝜑𝐴 ∈ Fin)
vdgrfiun.b (𝜑𝐵 ∈ Fin)
vdgrfiun.i (𝜑 → (𝐴𝐵) = ∅)
vdgrfiun.ge (𝜑𝑉 UMGrph 𝐸)
vdgrfiun.gf (𝜑𝑉 UMGrph 𝐹)
vdgrfiun.u (𝜑𝑈𝑉)
Assertion
Ref Expression
vdgrfiun (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)))

Proof of Theorem vdgrfiun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elun 3715 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 727 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)))
3 andir 908 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
42, 3bitri 263 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
54abbii 2726 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
6 df-rab 2905 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))}
7 unab 3853 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
85, 6, 73eqtr4i 2642 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))})
9 df-rab 2905 . . . . . . . . 9 {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))}
10 vdgrfiun.e . . . . . . . . . . . . 13 (𝜑𝐸 Fn 𝐴)
1110adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐸 Fn 𝐴)
12 vdgrfiun.f . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐵)
1312adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐹 Fn 𝐵)
14 vdgrfiun.i . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = ∅)
1514adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐴𝐵) = ∅)
16 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
17 fvun1 6179 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐴)) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1811, 13, 15, 16, 17syl112anc 1322 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1918eleq2d 2673 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐸𝑥)))
2019rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
219, 20syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
22 df-rab 2905 . . . . . . . . 9 {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}
2310adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐸 Fn 𝐴)
2412adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹 Fn 𝐵)
2514adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → (𝐴𝐵) = ∅)
26 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝑥𝐵)
27 fvun2 6180 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐵)) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2823, 24, 25, 26, 27syl112anc 1322 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2928eleq2d 2673 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐹𝑥)))
3029rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3122, 30syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3221, 31uneq12d 3730 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
338, 32syl5eq 2656 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
3433fveq2d 6107 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
35 vdgrfiun.a . . . . . . 7 (𝜑𝐴 ∈ Fin)
36 ssrab2 3650 . . . . . . 7 {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴
37 ssfi 8065 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴) → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin)
3835, 36, 37sylancl 693 . . . . . 6 (𝜑 → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin)
39 vdgrfiun.b . . . . . . 7 (𝜑𝐵 ∈ Fin)
40 ssrab2 3650 . . . . . . 7 {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵
41 ssfi 8065 . . . . . . 7 ((𝐵 ∈ Fin ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵) → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin)
4239, 40, 41sylancl 693 . . . . . 6 (𝜑 → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin)
43 ss2in 3802 . . . . . . . . 9 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴 ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵) → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵))
4436, 40, 43mp2an 704 . . . . . . . 8 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵)
4544, 14syl5sseq 3616 . . . . . . 7 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅)
46 ss0 3926 . . . . . . 7 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅ → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
4745, 46syl 17 . . . . . 6 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
48 hashun 13032 . . . . . 6 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin ∧ ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅) → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
4938, 42, 47, 48syl3anc 1318 . . . . 5 (𝜑 → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
5034, 49eqtrd 2644 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
511anbi1i 727 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))
52 andir 908 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5351, 52bitri 263 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5453abbii 2726 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
55 df-rab 2905 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
56 unab 3853 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
5754, 55, 563eqtr4i 2642 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})})
58 df-rab 2905 . . . . . . . . 9 {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
5918eqeq1d 2612 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐸𝑥) = {𝑈}))
6059rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
6158, 60syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
62 df-rab 2905 . . . . . . . . 9 {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
6328eqeq1d 2612 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐹𝑥) = {𝑈}))
6463rabbidva 3163 . . . . . . . . 9 (𝜑 → {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6562, 64syl5eqr 2658 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6661, 65uneq12d 3730 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6757, 66syl5eq 2656 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6867fveq2d 6107 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
69 ssrab2 3650 . . . . . . 7 {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴
70 ssfi 8065 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴) → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ Fin)
7135, 69, 70sylancl 693 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ Fin)
72 ssrab2 3650 . . . . . . 7 {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵
73 ssfi 8065 . . . . . . 7 ((𝐵 ∈ Fin ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵) → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ Fin)
7439, 72, 73sylancl 693 . . . . . 6 (𝜑 → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ Fin)
75 ss2in 3802 . . . . . . . . 9 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴 ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵) → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵))
7669, 72, 75mp2an 704 . . . . . . . 8 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵)
7776, 14syl5sseq 3616 . . . . . . 7 (𝜑 → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ ∅)
78 ss0 3926 . . . . . . 7 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ ∅ → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) = ∅)
7977, 78syl 17 . . . . . 6 (𝜑 → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) = ∅)
80 hashun 13032 . . . . . 6 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ Fin ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ Fin ∧ ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) = ∅) → (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8171, 74, 79, 80syl3anc 1318 . . . . 5 (𝜑 → (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8268, 81eqtrd 2644 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8350, 82oveq12d 6567 . . 3 (𝜑 → ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})) + ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
84 hashcl 13009 . . . . . 6 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℕ0)
8538, 84syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℕ0)
8685nn0cnd 11230 . . . 4 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℂ)
87 hashcl 13009 . . . . . 6 ({𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℕ0)
8842, 87syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℕ0)
8988nn0cnd 11230 . . . 4 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℂ)
90 hashcl 13009 . . . . . 6 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ Fin → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℕ0)
9171, 90syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℕ0)
9291nn0cnd 11230 . . . 4 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℂ)
93 hashcl 13009 . . . . . 6 ({𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ Fin → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℕ0)
9474, 93syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℕ0)
9594nn0cnd 11230 . . . 4 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℂ)
9686, 89, 92, 95add4d 10143 . . 3 (𝜑 → (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})) + ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) + ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
9783, 96eqtrd 2644 . 2 (𝜑 → ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) + ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
98 relumgra 25843 . . . 4 Rel UMGrph
99 vdgrfiun.ge . . . 4 (𝜑𝑉 UMGrph 𝐸)
100 brrelex 5080 . . . 4 ((Rel UMGrph ∧ 𝑉 UMGrph 𝐸) → 𝑉 ∈ V)
10198, 99, 100sylancr 694 . . 3 (𝜑𝑉 ∈ V)
102 fnun 5911 . . . 4 (((𝐸 Fn 𝐴𝐹 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐸𝐹) Fn (𝐴𝐵))
10310, 12, 14, 102syl21anc 1317 . . 3 (𝜑 → (𝐸𝐹) Fn (𝐴𝐵))
104 unfi 8112 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
10535, 39, 104syl2anc 691 . . 3 (𝜑 → (𝐴𝐵) ∈ Fin)
106 vdgrfiun.u . . 3 (𝜑𝑈𝑉)
107 vdgrfival 26424 . . 3 (((𝑉 ∈ V ∧ (𝐸𝐹) Fn (𝐴𝐵) ∧ (𝐴𝐵) ∈ Fin) ∧ 𝑈𝑉) → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})))
108101, 103, 105, 106, 107syl31anc 1321 . 2 (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})))
109 vdgrfival 26424 . . . 4 (((𝑉 ∈ V ∧ 𝐸 Fn 𝐴𝐴 ∈ Fin) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
110101, 10, 35, 106, 109syl31anc 1321 . . 3 (𝜑 → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
111 vdgrfival 26424 . . . 4 (((𝑉 ∈ V ∧ 𝐹 Fn 𝐵𝐵 ∈ Fin) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
112101, 12, 39, 106, 111syl31anc 1321 . . 3 (𝜑 → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
113110, 112oveq12d 6567 . 2 (𝜑 → (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) + ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
11497, 108, 1133eqtr4d 2654 1 (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘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
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