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Theorem vtxdun 40695
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 2901 . . . . . . . 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 5232 . . . . . . . . . . . . . 14 (𝜑 → dom (iEdg‘𝑈) = dom (𝐼𝐽))
4 dmun 5237 . . . . . . . . . . . . . 14 dom (𝐼𝐽) = (dom 𝐼 ∪ dom 𝐽)
53, 4syl6eq 2656 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽))
65eleq2d 2669 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽)))
7 elun 3711 . . . . . . . . . . . 12 (𝑥 ∈ (dom 𝐼 ∪ dom 𝐽) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽))
86, 7syl6bb 274 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽)))
98anbi1d 736 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
10 andir 907 . . . . . . . . . 10 (((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
119, 10syl6bb 274 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))))
1211abbidv 2724 . . . . . . . 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 2652 . . . . . . 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 3849 . . . . . . . . 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 2615 . . . . . . . 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 2901 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
182fveq1d 6087 . . . . . . . . . . . . 13 (𝜑 → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
1918adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
20 vtxdun.fi . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐼)
21 funfn 5816 . . . . . . . . . . . . . . 15 (Fun 𝐼𝐼 Fn dom 𝐼)
2220, 21sylib 206 . . . . . . . . . . . . . 14 (𝜑𝐼 Fn dom 𝐼)
2322adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
24 vtxdun.fj . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐽)
25 funfn 5816 . . . . . . . . . . . . . . 15 (Fun 𝐽𝐽 Fn dom 𝐽)
2624, 25sylib 206 . . . . . . . . . . . . . 14 (𝜑𝐽 Fn dom 𝐽)
2726adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐽 Fn dom 𝐽)
28 vtxdun.d . . . . . . . . . . . . . 14 (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
2928anim1i 589 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼))
30 fvun1 6161 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
3123, 27, 29, 30syl3anc 1317 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
3219, 31eqtrd 2640 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼𝑥))
3332eleq2d 2669 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼𝑥)))
3433rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
3517, 34syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
36 df-rab 2901 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
3718adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
3822adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐼 Fn dom 𝐼)
3926adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐽 Fn dom 𝐽)
4028anim1i 589 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽))
41 fvun2 6162 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4238, 39, 40, 41syl3anc 1317 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4337, 42eqtrd 2640 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽𝑥))
4443eleq2d 2669 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽𝑥)))
4544rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4636, 45syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4735, 46uneq12d 3726 . . . . . . 7 (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}))
4813, 16, 473eqtrd 2644 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}))
4948fveq2d 6089 . . . . 5 (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = (#‘({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
50 vtxdun.i . . . . . . . . . 10 𝐼 = (iEdg‘𝐺)
51 fvex 6095 . . . . . . . . . 10 (iEdg‘𝐺) ∈ V
5250, 51eqeltri 2680 . . . . . . . . 9 𝐼 ∈ V
5352dmex 6965 . . . . . . . 8 dom 𝐼 ∈ V
5453rabex 4732 . . . . . . 7 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V
5554a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V)
56 vtxdun.j . . . . . . . . . 10 𝐽 = (iEdg‘𝐻)
57 fvex 6095 . . . . . . . . . 10 (iEdg‘𝐻) ∈ V
5856, 57eqeltri 2680 . . . . . . . . 9 𝐽 ∈ V
5958dmex 6965 . . . . . . . 8 dom 𝐽 ∈ V
6059rabex 4732 . . . . . . 7 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V
6160a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V)
62 ssrab2 3646 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ⊆ dom 𝐼
63 ssrab2 3646 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ⊆ dom 𝐽
64 ss2in 3798 . . . . . . . . 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))
6562, 63, 64mp2an 703 . . . . . . . 8 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)
6665, 28syl5sseq 3612 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ ∅)
67 ss0 3922 . . . . . . 7 (({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) = ∅)
6866, 67syl 17 . . . . . 6 (𝜑 → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) = ∅)
69 hashunx 12985 . . . . . 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)})))
7055, 61, 68, 69syl3anc 1317 . . . . 5 (𝜑 → (#‘({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})) = ((#‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
7149, 70eqtrd 2640 . . . 4 (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = ((#‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
72 df-rab 2901 . . . . . . . 8 {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
738anbi1d 736 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
74 andir 907 . . . . . . . . . 10 (((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
7573, 74syl6bb 274 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))))
7675abbidv 2724 . . . . . . . 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}))})
7772, 76syl5eq 2652 . . . . . . 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}))})
78 unab 3849 . . . . . . . . 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}))}
7978eqcomi 2615 . . . . . . . 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})})
8079a1i 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})}))
81 df-rab 2901 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
8232eqeq1d 2608 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼𝑥) = {𝑁}))
8382rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
8481, 83syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
85 df-rab 2901 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
8643eqeq1d 2608 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽𝑥) = {𝑁}))
8786rabbidva 3159 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8885, 87syl5eqr 2654 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8984, 88uneq12d 3726 . . . . . . 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}}))
9077, 80, 893eqtrd 2644 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))
9190fveq2d 6089 . . . . 5 (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = (#‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
9253rabex 4732 . . . . . . 7 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V
9392a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V)
9459rabex 4732 . . . . . . 7 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V
9594a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V)
96 ssrab2 3646 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ⊆ dom 𝐼
97 ssrab2 3646 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ⊆ dom 𝐽
98 ss2in 3798 . . . . . . . . 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))
9996, 97, 98mp2an 703 . . . . . . . 8 ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)
10099, 28syl5sseq 3612 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ ∅)
101 ss0 3922 . . . . . . 7 (({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) = ∅)
102100, 101syl 17 . . . . . 6 (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) = ∅)
103 hashunx 12985 . . . . . 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}})))
10493, 95, 102, 103syl3anc 1317 . . . . 5 (𝜑 → (#‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})) = ((#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10591, 104eqtrd 2640 . . . 4 (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = ((#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10671, 105oveq12d 6542 . . 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}}))))
107 hashxnn0 40217 . . . . 5 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V → (#‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
10855, 107syl 17 . . . 4 (𝜑 → (#‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
109 hashxnn0 40217 . . . . 5 ({𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V → (#‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
11061, 109syl 17 . . . 4 (𝜑 → (#‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
111 hashxnn0 40217 . . . . 5 ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V → (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) ∈ ℕ0*)
11293, 111syl 17 . . . 4 (𝜑 → (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) ∈ ℕ0*)
113 hashxnn0 40217 . . . . 5 ({𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V → (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ∈ ℕ0*)
11495, 113syl 17 . . . 4 (𝜑 → (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ∈ ℕ0*)
115108, 110, 112, 114xnn0add4d 40213 . . 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}}))))
116106, 115eqtrd 2640 . 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}}))))
117 vtxdun.n . . . 4 (𝜑𝑁𝑉)
118 vtxdun.vu . . . 4 (𝜑 → (Vtx‘𝑈) = 𝑉)
119117, 118eleqtrrd 2687 . . 3 (𝜑𝑁 ∈ (Vtx‘𝑈))
120 eqid 2606 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
121 eqid 2606 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
122 eqid 2606 . . . 4 dom (iEdg‘𝑈) = dom (iEdg‘𝑈)
123120, 121, 122vtxdgval 40683 . . 3 (𝑁 ∈ (Vtx‘𝑈) → ((VtxDeg‘𝑈)‘𝑁) = ((#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
124119, 123syl 17 . 2 (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = ((#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
125 vtxdun.vg . . . . 5 𝑉 = (Vtx‘𝐺)
126 eqid 2606 . . . . 5 dom 𝐼 = dom 𝐼
127125, 50, 126vtxdgval 40683 . . . 4 (𝑁𝑉 → ((VtxDeg‘𝐺)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})))
128117, 127syl 17 . . 3 (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})))
129 vtxdun.vh . . . . 5 (𝜑 → (Vtx‘𝐻) = 𝑉)
130117, 129eleqtrrd 2687 . . . 4 (𝜑𝑁 ∈ (Vtx‘𝐻))
131 eqid 2606 . . . . 5 (Vtx‘𝐻) = (Vtx‘𝐻)
132 eqid 2606 . . . . 5 dom 𝐽 = dom 𝐽
133131, 56, 132vtxdgval 40683 . . . 4 (𝑁 ∈ (Vtx‘𝐻) → ((VtxDeg‘𝐻)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
134130, 133syl 17 . . 3 (𝜑 → ((VtxDeg‘𝐻)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
135128, 134oveq12d 6542 . 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}}))))
136116, 124, 1353eqtr4d 2650 1 (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
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  dom cdm 5025  Fun wfun 5781   Fn wfn 5782  cfv 5787  (class class class)co 6524   +𝑒 cxad 11773  #chash 12931  0*cxnn0 40196  Vtxcvtx 40228  iEdgciedg 40229  VtxDegcvtxdg 40680
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-1st 7033  df-2nd 7034  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-xnn0 40197  df-vtxdg 40681
This theorem is referenced by:  vtxdfiun  40696  vtxduhgrun  40697  p1evtxdeqlem  40727
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