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Theorem vtxdun 27190
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 3144 . . . . . . . 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 5767 . . . . . . . . . . . . . 14 (𝜑 → dom (iEdg‘𝑈) = dom (𝐼𝐽))
4 dmun 5772 . . . . . . . . . . . . . 14 dom (𝐼𝐽) = (dom 𝐼 ∪ dom 𝐽)
53, 4syl6eq 2869 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽))
65eleq2d 2895 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽)))
7 elun 4122 . . . . . . . . . . . 12 (𝑥 ∈ (dom 𝐼 ∪ dom 𝐽) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽))
86, 7syl6bb 288 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽)))
98anbi1d 629 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
10 andir 1002 . . . . . . . . . 10 (((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
119, 10syl6bb 288 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))))
1211abbidv 2882 . . . . . . . 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 2865 . . . . . . 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 4267 . . . . . . . . 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 2827 . . . . . . . 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 3144 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
182fveq1d 6665 . . . . . . . . . . . . 13 (𝜑 → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
1918adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
20 vtxdun.fi . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐼)
2120funfnd 6379 . . . . . . . . . . . . . 14 (𝜑𝐼 Fn dom 𝐼)
2221adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
23 vtxdun.fj . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐽)
2423funfnd 6379 . . . . . . . . . . . . . 14 (𝜑𝐽 Fn dom 𝐽)
2524adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐽 Fn dom 𝐽)
26 vtxdun.d . . . . . . . . . . . . . 14 (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
2726anim1i 614 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼))
28 fvun1 6747 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
2922, 25, 27, 28syl3anc 1363 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
3019, 29eqtrd 2853 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼𝑥))
3130eleq2d 2895 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼𝑥)))
3231rabbidva 3476 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
3317, 32syl5eqr 2867 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
34 df-rab 3144 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
3518adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
3621adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐼 Fn dom 𝐼)
3724adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐽 Fn dom 𝐽)
3826anim1i 614 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽))
39 fvun2 6748 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4036, 37, 38, 39syl3anc 1363 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4135, 40eqtrd 2853 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽𝑥))
4241eleq2d 2895 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽𝑥)))
4342rabbidva 3476 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4434, 43syl5eqr 2867 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4533, 44uneq12d 4137 . . . . . . 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 2857 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}))
4746fveq2d 6667 . . . . 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 6677 . . . . . . . . 9 𝐼 ∈ V
5049dmex 7605 . . . . . . . 8 dom 𝐼 ∈ V
5150rabex 5226 . . . . . . 7 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V
5251a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V)
53 vtxdun.j . . . . . . . . . 10 𝐽 = (iEdg‘𝐻)
5453fvexi 6677 . . . . . . . . 9 𝐽 ∈ V
5554dmex 7605 . . . . . . . 8 dom 𝐽 ∈ V
5655rabex 5226 . . . . . . 7 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V
5756a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V)
58 ssrab2 4053 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ⊆ dom 𝐼
59 ssrab2 4053 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ⊆ dom 𝐽
60 ss2in 4210 . . . . . . . . 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 688 . . . . . . . 8 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)
6261, 26sseqtrid 4016 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ ∅)
63 ss0 4349 . . . . . . 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 13735 . . . . . 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 1363 . . . . 5 (𝜑 → (♯‘({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
6747, 66eqtrd 2853 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
68 df-rab 3144 . . . . . . . 8 {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
698anbi1d 629 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
70 andir 1002 . . . . . . . . . 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 288 . . . . . . . . 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 2882 . . . . . . . 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 2865 . . . . . . 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 4267 . . . . . . . . 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 2827 . . . . . . . 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 3144 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
7830eqeq1d 2820 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼𝑥) = {𝑁}))
7978rabbidva 3476 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
8077, 79syl5eqr 2867 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
81 df-rab 3144 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
8241eqeq1d 2820 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽𝑥) = {𝑁}))
8382rabbidva 3476 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8481, 83syl5eqr 2867 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8580, 84uneq12d 4137 . . . . . . 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 2857 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))
8786fveq2d 6667 . . . . 5 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
8850rabex 5226 . . . . . . 7 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V
8988a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V)
9055rabex 5226 . . . . . . 7 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V
9190a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V)
92 ssrab2 4053 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ⊆ dom 𝐼
93 ssrab2 4053 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ⊆ dom 𝐽
94 ss2in 4210 . . . . . . . . 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 688 . . . . . . . 8 ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)
9695, 26sseqtrid 4016 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ ∅)
97 ss0 4349 . . . . . . 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 13735 . . . . . 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 1363 . . . . 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 2853 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10267, 101oveq12d 7163 . . 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 13687 . . . . 5 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
10452, 103syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
105 hashxnn0 13687 . . . . 5 ({𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
10657, 105syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
107 hashxnn0 13687 . . . . 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 13687 . . . . 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 12685 . . 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 2853 . 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 2913 . . 3 (𝜑𝑁 ∈ (Vtx‘𝑈))
116 eqid 2818 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
117 eqid 2818 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
118 eqid 2818 . . . 4 dom (iEdg‘𝑈) = dom (iEdg‘𝑈)
119116, 117, 118vtxdgval 27177 . . 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 2818 . . . . 5 dom 𝐼 = dom 𝐼
123121, 48, 122vtxdgval 27177 . . . 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 2913 . . . 4 (𝜑𝑁 ∈ (Vtx‘𝐻))
127 eqid 2818 . . . . 5 (Vtx‘𝐻) = (Vtx‘𝐻)
128 eqid 2818 . . . . 5 dom 𝐽 = dom 𝐽
129127, 53, 128vtxdgval 27177 . . . 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 7163 . 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 2863 1 (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  {cab 2796  {crab 3139  Vcvv 3492  cun 3931  cin 3932  wss 3933  c0 4288  {csn 4557  dom cdm 5548  Fun wfun 6342   Fn wfn 6343  cfv 6348  (class class class)co 7145  0*cxnn0 11955   +𝑒 cxad 12493  chash 13678  Vtxcvtx 26708  iEdgciedg 26709  VtxDegcvtxdg 27174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-xadd 12496  df-hash 13679  df-vtxdg 27175
This theorem is referenced by:  vtxdfiun  27191  vtxduhgrun  27192  p1evtxdeqlem  27221
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