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Theorem vtxdun 27265
Description: The degree of a vertex in the union of two graphs 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-Dec-2017.) (Revised by AV, 19-Feb-2021.)
Hypotheses
Ref Expression
vtxdun.i 𝐼 = (iEdg‘𝐺)
vtxdun.j 𝐽 = (iEdg‘𝐻)
vtxdun.vg 𝑉 = (Vtx‘𝐺)
vtxdun.vh (𝜑 → (Vtx‘𝐻) = 𝑉)
vtxdun.vu (𝜑 → (Vtx‘𝑈) = 𝑉)
vtxdun.d (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
vtxdun.fi (𝜑 → Fun 𝐼)
vtxdun.fj (𝜑 → Fun 𝐽)
vtxdun.n (𝜑𝑁𝑉)
vtxdun.u (𝜑 → (iEdg‘𝑈) = (𝐼𝐽))
Assertion
Ref Expression
vtxdun (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))

Proof of Theorem vtxdun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-rab 3149 . . . . . . . 8 {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
2 vtxdun.u . . . . . . . . . . . . . . 15 (𝜑 → (iEdg‘𝑈) = (𝐼𝐽))
32dmeqd 5776 . . . . . . . . . . . . . 14 (𝜑 → dom (iEdg‘𝑈) = dom (𝐼𝐽))
4 dmun 5781 . . . . . . . . . . . . . 14 dom (𝐼𝐽) = (dom 𝐼 ∪ dom 𝐽)
53, 4syl6eq 2874 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽))
65eleq2d 2900 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽)))
7 elun 4127 . . . . . . . . . . . 12 (𝑥 ∈ (dom 𝐼 ∪ dom 𝐽) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽))
86, 7syl6bb 289 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽)))
98anbi1d 631 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
10 andir 1005 . . . . . . . . . 10 (((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
119, 10syl6bb 289 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))))
1211abbidv 2887 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))})
131, 12syl5eq 2870 . . . . . . 7 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))})
14 unab 4272 . . . . . . . . 9 ({𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = {𝑥 ∣ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))}
1514eqcomi 2832 . . . . . . . 8 {𝑥 ∣ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))})
1615a1i 11 . . . . . . 7 (𝜑 → {𝑥 ∣ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}))
17 df-rab 3149 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
182fveq1d 6674 . . . . . . . . . . . . 13 (𝜑 → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
1918adantr 483 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
20 vtxdun.fi . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐼)
2120funfnd 6388 . . . . . . . . . . . . . 14 (𝜑𝐼 Fn dom 𝐼)
2221adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
23 vtxdun.fj . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐽)
2423funfnd 6388 . . . . . . . . . . . . . 14 (𝜑𝐽 Fn dom 𝐽)
2524adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐽 Fn dom 𝐽)
26 vtxdun.d . . . . . . . . . . . . . 14 (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
2726anim1i 616 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼))
28 fvun1 6756 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
2922, 25, 27, 28syl3anc 1367 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
3019, 29eqtrd 2858 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼𝑥))
3130eleq2d 2900 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼𝑥)))
3231rabbidva 3480 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
3317, 32syl5eqr 2872 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
34 df-rab 3149 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
3518adantr 483 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
3621adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐼 Fn dom 𝐼)
3724adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐽 Fn dom 𝐽)
3826anim1i 616 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽))
39 fvun2 6757 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4036, 37, 38, 39syl3anc 1367 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4135, 40eqtrd 2858 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽𝑥))
4241eleq2d 2900 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽𝑥)))
4342rabbidva 3480 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4434, 43syl5eqr 2872 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4533, 44uneq12d 4142 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}))
4613, 16, 453eqtrd 2862 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}))
4746fveq2d 6676 . . . . 5 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = (♯‘({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
48 vtxdun.i . . . . . . . . . 10 𝐼 = (iEdg‘𝐺)
4948fvexi 6686 . . . . . . . . 9 𝐼 ∈ V
5049dmex 7618 . . . . . . . 8 dom 𝐼 ∈ V
5150rabex 5237 . . . . . . 7 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V
5251a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V)
53 vtxdun.j . . . . . . . . . 10 𝐽 = (iEdg‘𝐻)
5453fvexi 6686 . . . . . . . . 9 𝐽 ∈ V
5554dmex 7618 . . . . . . . 8 dom 𝐽 ∈ V
5655rabex 5237 . . . . . . 7 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V
5756a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V)
58 ssrab2 4058 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ⊆ dom 𝐼
59 ssrab2 4058 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ⊆ dom 𝐽
60 ss2in 4215 . . . . . . . . 9 (({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽))
6158, 59, 60mp2an 690 . . . . . . . 8 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)
6261, 26sseqtrid 4021 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ ∅)
63 ss0 4354 . . . . . . 7 (({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) = ∅)
6462, 63syl 17 . . . . . 6 (𝜑 → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) = ∅)
65 hashunx 13750 . . . . . 6 (({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V ∧ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V ∧ ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) = ∅) → (♯‘({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
6652, 57, 64, 65syl3anc 1367 . . . . 5 (𝜑 → (♯‘({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
6747, 66eqtrd 2858 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
68 df-rab 3149 . . . . . . . 8 {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
698anbi1d 631 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
70 andir 1005 . . . . . . . . . 10 (((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
7169, 70syl6bb 289 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))))
7271abbidv 2887 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))})
7368, 72syl5eq 2870 . . . . . . 7 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))})
74 unab 4272 . . . . . . . . 9 ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}) = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))}
7574eqcomi 2832 . . . . . . . 8 {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})})
7675a1i 11 . . . . . . 7 (𝜑 → {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}))
77 df-rab 3149 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
7830eqeq1d 2825 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼𝑥) = {𝑁}))
7978rabbidva 3480 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
8077, 79syl5eqr 2872 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
81 df-rab 3149 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
8241eqeq1d 2825 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽𝑥) = {𝑁}))
8382rabbidva 3480 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8481, 83syl5eqr 2872 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8580, 84uneq12d 4142 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}) = ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))
8673, 76, 853eqtrd 2862 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))
8786fveq2d 6676 . . . . 5 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
8850rabex 5237 . . . . . . 7 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V
8988a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V)
9055rabex 5237 . . . . . . 7 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V
9190a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V)
92 ssrab2 4058 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ⊆ dom 𝐼
93 ssrab2 4058 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ⊆ dom 𝐽
94 ss2in 4215 . . . . . . . . 9 (({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽))
9592, 93, 94mp2an 690 . . . . . . . 8 ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)
9695, 26sseqtrid 4021 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ ∅)
97 ss0 4354 . . . . . . 7 (({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) = ∅)
9896, 97syl 17 . . . . . 6 (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) = ∅)
99 hashunx 13750 . . . . . 6 (({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V ∧ ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) = ∅) → (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10089, 91, 98, 99syl3anc 1367 . . . . 5 (𝜑 → (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10187, 100eqtrd 2858 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10267, 101oveq12d 7176 . . 3 (𝜑 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})) +𝑒 ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))))
103 hashxnn0 13702 . . . . 5 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
10452, 103syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
105 hashxnn0 13702 . . . . 5 ({𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
10657, 105syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
107 hashxnn0 13702 . . . . 5 ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) ∈ ℕ0*)
10889, 107syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) ∈ ℕ0*)
109 hashxnn0 13702 . . . . 5 ({𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ∈ ℕ0*)
11091, 109syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ∈ ℕ0*)
111104, 106, 108, 110xnn0add4d 12700 . . 3 (𝜑 → (((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})) +𝑒 ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))) = (((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})) +𝑒 ((♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))))
112102, 111eqtrd 2858 . 2 (𝜑 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})) +𝑒 ((♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))))
113 vtxdun.n . . . 4 (𝜑𝑁𝑉)
114 vtxdun.vu . . . 4 (𝜑 → (Vtx‘𝑈) = 𝑉)
115113, 114eleqtrrd 2918 . . 3 (𝜑𝑁 ∈ (Vtx‘𝑈))
116 eqid 2823 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
117 eqid 2823 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
118 eqid 2823 . . . 4 dom (iEdg‘𝑈) = dom (iEdg‘𝑈)
119116, 117, 118vtxdgval 27252 . . 3 (𝑁 ∈ (Vtx‘𝑈) → ((VtxDeg‘𝑈)‘𝑁) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
120115, 119syl 17 . 2 (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
121 vtxdun.vg . . . . 5 𝑉 = (Vtx‘𝐺)
122 eqid 2823 . . . . 5 dom 𝐼 = dom 𝐼
123121, 48, 122vtxdgval 27252 . . . 4 (𝑁𝑉 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})))
124113, 123syl 17 . . 3 (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})))
125 vtxdun.vh . . . . 5 (𝜑 → (Vtx‘𝐻) = 𝑉)
126113, 125eleqtrrd 2918 . . . 4 (𝜑𝑁 ∈ (Vtx‘𝐻))
127 eqid 2823 . . . . 5 (Vtx‘𝐻) = (Vtx‘𝐻)
128 eqid 2823 . . . . 5 dom 𝐽 = dom 𝐽
129127, 53, 128vtxdgval 27252 . . . 4 (𝑁 ∈ (Vtx‘𝐻) → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
130126, 129syl 17 . . 3 (𝜑 → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
131124, 130oveq12d 7176 . 2 (𝜑 → (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)) = (((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})) +𝑒 ((♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))))
132112, 120, 1313eqtr4d 2868 1 (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843   = wceq 1537  wcel 2114  {cab 2801  {crab 3144  Vcvv 3496  cun 3936  cin 3937  wss 3938  c0 4293  {csn 4569  dom cdm 5557  Fun wfun 6351   Fn wfn 6352  cfv 6357  (class class class)co 7158  0*cxnn0 11970   +𝑒 cxad 12508  chash 13693  Vtxcvtx 26783  iEdgciedg 26784  VtxDegcvtxdg 27249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-xadd 12511  df-hash 13694  df-vtxdg 27250
This theorem is referenced by:  vtxdfiun  27266  vtxduhgrun  27267  p1evtxdeqlem  27296
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