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Theorem vdgrfiun 26192
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 3711 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 726 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)))
3 andir 907 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
42, 3bitri 262 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥)) ↔ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))))
54abbii 2722 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
6 df-rab 2901 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑈 ∈ ((𝐸𝐹)‘𝑥))}
7 unab 3849 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = {𝑥 ∣ ((𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)) ∨ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)))}
85, 6, 73eqtr4i 2638 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))})
9 df-rab 2901 . . . . . . . . 9 {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))}
10 vdgrfiun.e . . . . . . . . . . . . 13 (𝜑𝐸 Fn 𝐴)
1110adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐸 Fn 𝐴)
12 vdgrfiun.f . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝐵)
1312adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐹 Fn 𝐵)
14 vdgrfiun.i . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐵) = ∅)
1514adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐴𝐵) = ∅)
16 simpr 475 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝑥𝐴)
17 fvun1 6161 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐴)) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1811, 13, 15, 16, 17syl112anc 1321 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐸𝐹)‘𝑥) = (𝐸𝑥))
1918eleq2d 2669 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐸𝑥)))
2019rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
219, 20syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐴𝑈 ∈ (𝐸𝑥)})
22 df-rab 2901 . . . . . . . . 9 {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}
2310adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐸 Fn 𝐴)
2412adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹 Fn 𝐵)
2514adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → (𝐴𝐵) = ∅)
26 simpr 475 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝑥𝐵)
27 fvun2 6162 . . . . . . . . . . . 12 ((𝐸 Fn 𝐴𝐹 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑥𝐵)) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2823, 24, 25, 26, 27syl112anc 1321 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((𝐸𝐹)‘𝑥) = (𝐹𝑥))
2928eleq2d 2669 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (𝑈 ∈ ((𝐸𝐹)‘𝑥) ↔ 𝑈 ∈ (𝐹𝑥)))
3029rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥)} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3122, 30syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))} = {𝑥𝐵𝑈 ∈ (𝐹𝑥)})
3221, 31uneq12d 3726 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴𝑈 ∈ ((𝐸𝐹)‘𝑥))} ∪ {𝑥 ∣ (𝑥𝐵𝑈 ∈ ((𝐸𝐹)‘𝑥))}) = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
338, 32syl5eq 2652 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)} = ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}))
3433fveq2d 6089 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
35 vdgrfiun.a . . . . . . 7 (𝜑𝐴 ∈ Fin)
36 ssrab2 3646 . . . . . . 7 {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴
37 ssfi 8039 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴) → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin)
3835, 36, 37sylancl 692 . . . . . 6 (𝜑 → {𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin)
39 vdgrfiun.b . . . . . . 7 (𝜑𝐵 ∈ Fin)
40 ssrab2 3646 . . . . . . 7 {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵
41 ssfi 8039 . . . . . . 7 ((𝐵 ∈ Fin ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵) → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin)
4239, 40, 41sylancl 692 . . . . . 6 (𝜑 → {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin)
43 ss2in 3798 . . . . . . . . 9 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ⊆ 𝐴 ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ⊆ 𝐵) → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵))
4436, 40, 43mp2an 703 . . . . . . . 8 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ (𝐴𝐵)
4544, 14syl5sseq 3612 . . . . . . 7 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅)
46 ss0 3922 . . . . . . 7 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ⊆ ∅ → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
4745, 46syl 17 . . . . . 6 (𝜑 → ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅)
48 hashun 12981 . . . . . 6 (({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin ∧ {𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin ∧ ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∩ {𝑥𝐵𝑈 ∈ (𝐹𝑥)}) = ∅) → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
4938, 42, 47, 48syl3anc 1317 . . . . 5 (𝜑 → (#‘({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∪ {𝑥𝐵𝑈 ∈ (𝐹𝑥)})) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
5034, 49eqtrd 2640 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})))
511anbi1i 726 . . . . . . . . . 10 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))
52 andir 907 . . . . . . . . . 10 (((𝑥𝐴𝑥𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5351, 52bitri 262 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ↔ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})))
5453abbii 2722 . . . . . . . 8 {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
55 df-rab 2901 . . . . . . . 8 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥 ∈ (𝐴𝐵) ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
56 unab 3849 . . . . . . . 8 ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = {𝑥 ∣ ((𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}) ∨ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈}))}
5754, 55, 563eqtr4i 2638 . . . . . . 7 {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})})
58 df-rab 2901 . . . . . . . . 9 {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
5918eqeq1d 2608 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐸𝑥) = {𝑈}))
6059rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐴 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
6158, 60syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})
62 df-rab 2901 . . . . . . . . 9 {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}
6328eqeq1d 2608 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (((𝐸𝐹)‘𝑥) = {𝑈} ↔ (𝐹𝑥) = {𝑈}))
6463rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥𝐵 ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6562, 64syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} = {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})
6661, 65uneq12d 3726 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥𝐴 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})} ∪ {𝑥 ∣ (𝑥𝐵 ∧ ((𝐸𝐹)‘𝑥) = {𝑈})}) = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6757, 66syl5eq 2652 . . . . . 6 (𝜑 → {𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}} = ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))
6867fveq2d 6089 . . . . 5 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
69 ssrab2 3646 . . . . . . 7 {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴
70 ssfi 8039 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴) → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ Fin)
7135, 69, 70sylancl 692 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ Fin)
72 ssrab2 3646 . . . . . . 7 {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵
73 ssfi 8039 . . . . . . 7 ((𝐵 ∈ Fin ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵) → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ Fin)
7439, 72, 73sylancl 692 . . . . . 6 (𝜑 → {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ Fin)
75 ss2in 3798 . . . . . . . . 9 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ⊆ 𝐴 ∧ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ⊆ 𝐵) → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵))
7669, 72, 75mp2an 703 . . . . . . . 8 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ (𝐴𝐵)
7776, 14syl5sseq 3612 . . . . . . 7 (𝜑 → ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∩ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ⊆ ∅)
78 ss0 3922 . . . . . . 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 12981 . . . . . 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 1317 . . . . 5 (𝜑 → (#‘({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∪ {𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8268, 81eqtrd 2640 . . . 4 (𝜑 → (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}}) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
8350, 82oveq12d 6542 . . 3 (𝜑 → ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})) + ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
84 hashcl 12958 . . . . . 6 ({𝑥𝐴𝑈 ∈ (𝐸𝑥)} ∈ Fin → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℕ0)
8538, 84syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℕ0)
8685nn0cnd 11197 . . . 4 (𝜑 → (#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) ∈ ℂ)
87 hashcl 12958 . . . . . 6 ({𝑥𝐵𝑈 ∈ (𝐹𝑥)} ∈ Fin → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℕ0)
8842, 87syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℕ0)
8988nn0cnd 11197 . . . 4 (𝜑 → (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) ∈ ℂ)
90 hashcl 12958 . . . . . 6 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}} ∈ Fin → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℕ0)
9171, 90syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℕ0)
9291nn0cnd 11197 . . . 4 (𝜑 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) ∈ ℂ)
93 hashcl 12958 . . . . . 6 ({𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}} ∈ Fin → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℕ0)
9474, 93syl 17 . . . . 5 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℕ0)
9594nn0cnd 11197 . . . 4 (𝜑 → (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}) ∈ ℂ)
9686, 89, 92, 95add4d 10112 . . 3 (𝜑 → (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)})) + ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) + ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
9783, 96eqtrd 2640 . 2 (𝜑 → ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) + ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
98 relumgra 25606 . . . 4 Rel UMGrph
99 vdgrfiun.ge . . . 4 (𝜑𝑉 UMGrph 𝐸)
100 brrelex 5067 . . . 4 ((Rel UMGrph ∧ 𝑉 UMGrph 𝐸) → 𝑉 ∈ V)
10198, 99, 100sylancr 693 . . 3 (𝜑𝑉 ∈ V)
102 fnun 5894 . . . 4 (((𝐸 Fn 𝐴𝐹 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐸𝐹) Fn (𝐴𝐵))
10310, 12, 14, 102syl21anc 1316 . . 3 (𝜑 → (𝐸𝐹) Fn (𝐴𝐵))
104 unfi 8086 . . . 4 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
10535, 39, 104syl2anc 690 . . 3 (𝜑 → (𝐴𝐵) ∈ Fin)
106 vdgrfiun.u . . 3 (𝜑𝑈𝑉)
107 vdgrfival 26187 . . 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 1320 . 2 (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = ((#‘{𝑥 ∈ (𝐴𝐵) ∣ 𝑈 ∈ ((𝐸𝐹)‘𝑥)}) + (#‘{𝑥 ∈ (𝐴𝐵) ∣ ((𝐸𝐹)‘𝑥) = {𝑈}})))
109 vdgrfival 26187 . . . 4 (((𝑉 ∈ V ∧ 𝐸 Fn 𝐴𝐴 ∈ Fin) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
110101, 10, 35, 106, 109syl31anc 1320 . . 3 (𝜑 → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))
111 vdgrfival 26187 . . . 4 (((𝑉 ∈ V ∧ 𝐹 Fn 𝐵𝐵 ∈ Fin) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
112101, 12, 39, 106, 111syl31anc 1320 . . 3 (𝜑 → ((𝑉 VDeg 𝐹)‘𝑈) = ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}})))
113110, 112oveq12d 6542 . 2 (𝜑 → (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)) = (((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})) + ((#‘{𝑥𝐵𝑈 ∈ (𝐹𝑥)}) + (#‘{𝑥𝐵 ∣ (𝐹𝑥) = {𝑈}}))))
11497, 108, 1133eqtr4d 2650 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 381  wa 382   = wceq 1474  wcel 1976  {cab 2592  {crab 2896  Vcvv 3169  cun 3534  cin 3535  wss 3536  c0 3870  {csn 4121   class class class wbr 4574  Rel wrel 5030   Fn wfn 5782  cfv 5787  (class class class)co 6524  Fincfn 7815   + caddc 9792  0cn0 11136  #chash 12931   UMGrph cumg 25604   VDeg cvdg 26183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-n0 11137  df-z 11208  df-uz 11517  df-xadd 11776  df-hash 12932  df-umgra 25605  df-vdgr 26184
This theorem is referenced by:  eupath2lem3  26269  vdegp1ai  26274  vdegp1bi  26275
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