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Theorem vtxdun 26723
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 3096 . . . . . . . 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 5527 . . . . . . . . . . . . . 14 (𝜑 → dom (iEdg‘𝑈) = dom (𝐼𝐽))
4 dmun 5532 . . . . . . . . . . . . . 14 dom (𝐼𝐽) = (dom 𝐼 ∪ dom 𝐽)
53, 4syl6eq 2847 . . . . . . . . . . . . 13 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽))
65eleq2d 2862 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽)))
7 elun 3949 . . . . . . . . . . . 12 (𝑥 ∈ (dom 𝐼 ∪ dom 𝐽) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽))
86, 7syl6bb 279 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ (𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽)))
98anbi1d 624 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
10 andir 1032 . . . . . . . . . 10 (((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
119, 10syl6bb 279 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))))
1211abbidv 2916 . . . . . . . 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 2843 . . . . . . 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 4092 . . . . . . . . 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 2806 . . . . . . . 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 3096 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
182fveq1d 6411 . . . . . . . . . . . . 13 (𝜑 → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
1918adantr 473 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
20 vtxdun.fi . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐼)
21 funfn 6129 . . . . . . . . . . . . . . 15 (Fun 𝐼𝐼 Fn dom 𝐼)
2220, 21sylib 210 . . . . . . . . . . . . . 14 (𝜑𝐼 Fn dom 𝐼)
2322adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
24 vtxdun.fj . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝐽)
25 funfn 6129 . . . . . . . . . . . . . . 15 (Fun 𝐽𝐽 Fn dom 𝐽)
2624, 25sylib 210 . . . . . . . . . . . . . 14 (𝜑𝐽 Fn dom 𝐽)
2726adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → 𝐽 Fn dom 𝐽)
28 vtxdun.d . . . . . . . . . . . . . 14 (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
2928anim1i 609 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐼) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼))
30 fvun1 6492 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
3123, 27, 29, 30syl3anc 1491 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐼) → ((𝐼𝐽)‘𝑥) = (𝐼𝑥))
3219, 31eqtrd 2831 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼𝑥))
3332eleq2d 2862 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼𝑥)))
3433rabbidva 3370 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
3517, 34syl5eqr 2845 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)})
36 df-rab 3096 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
3718adantr 473 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = ((𝐼𝐽)‘𝑥))
3822adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐼 Fn dom 𝐼)
3926adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → 𝐽 Fn dom 𝐽)
4028anim1i 609 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ dom 𝐽) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽))
41 fvun2 6493 . . . . . . . . . . . . 13 ((𝐼 Fn dom 𝐼𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4238, 39, 40, 41syl3anc 1491 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ dom 𝐽) → ((𝐼𝐽)‘𝑥) = (𝐽𝑥))
4337, 42eqtrd 2831 . . . . . . . . . . 11 ((𝜑𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽𝑥))
4443eleq2d 2862 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽𝑥)))
4544rabbidva 3370 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4636, 45syl5eqr 2845 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})
4735, 46uneq12d 3964 . . . . . . 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 2835 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}))
4948fveq2d 6413 . . . . 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)
5150fvexi 6423 . . . . . . . . 9 𝐼 ∈ V
5251dmex 7332 . . . . . . . 8 dom 𝐼 ∈ V
5352rabex 5005 . . . . . . 7 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V
5453a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V)
55 vtxdun.j . . . . . . . . . 10 𝐽 = (iEdg‘𝐻)
5655fvexi 6423 . . . . . . . . 9 𝐽 ∈ V
5756dmex 7332 . . . . . . . 8 dom 𝐽 ∈ V
5857rabex 5005 . . . . . . 7 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V
5958a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V)
60 ssrab2 3881 . . . . . . . . 9 {𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ⊆ dom 𝐼
61 ssrab2 3881 . . . . . . . . 9 {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ⊆ dom 𝐽
62 ss2in 4034 . . . . . . . . 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))
6360, 61, 62mp2an 684 . . . . . . . 8 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)
6463, 28syl5sseq 3847 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ ∅)
65 ss0 4168 . . . . . . 7 (({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) = ∅)
6664, 65syl 17 . . . . . 6 (𝜑 → ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∩ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) = ∅)
67 hashunx 13421 . . . . . 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)})))
6854, 59, 66, 67syl3anc 1491 . . . . 5 (𝜑 → (♯‘({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∪ {𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
6949, 68eqtrd 2831 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)})))
70 df-rab 3096 . . . . . . . 8 {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
718anbi1d 624 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
72 andir 1032 . . . . . . . . . 10 (((𝑥 ∈ dom 𝐼𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
7371, 72syl6bb 279 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))))
7473abbidv 2916 . . . . . . . 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}))})
7570, 74syl5eq 2843 . . . . . . 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}))})
76 unab 4092 . . . . . . . . 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}))}
7776eqcomi 2806 . . . . . . . 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})})
7877a1i 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})}))
79 df-rab 3096 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
8032eqeq1d 2799 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼𝑥) = {𝑁}))
8180rabbidva 3370 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
8279, 81syl5eqr 2845 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})
83 df-rab 3096 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
8443eqeq1d 2799 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽𝑥) = {𝑁}))
8584rabbidva 3370 . . . . . . . . 9 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8683, 85syl5eqr 2845 . . . . . . . 8 (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})
8782, 86uneq12d 3964 . . . . . . 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}}))
8875, 78, 873eqtrd 2835 . . . . . 6 (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}))
8988fveq2d 6413 . . . . 5 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
9052rabex 5005 . . . . . . 7 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V
9190a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V)
9257rabex 5005 . . . . . . 7 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V
9392a1i 11 . . . . . 6 (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V)
94 ssrab2 3881 . . . . . . . . 9 {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ⊆ dom 𝐼
95 ssrab2 3881 . . . . . . . . 9 {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ⊆ dom 𝐽
96 ss2in 4034 . . . . . . . . 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))
9794, 95, 96mp2an 684 . . . . . . . 8 ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)
9897, 28syl5sseq 3847 . . . . . . 7 (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ ∅)
99 ss0 4168 . . . . . . 7 (({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) = ∅)
10098, 99syl 17 . . . . . 6 (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) = ∅)
101 hashunx 13421 . . . . . 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}})))
10291, 93, 100, 101syl3anc 1491 . . . . 5 (𝜑 → (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10389, 102eqtrd 2831 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
10469, 103oveq12d 6894 . . 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}}))))
105 hashxnn0 13375 . . . . 5 ({𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
10654, 105syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) ∈ ℕ0*)
107 hashxnn0 13375 . . . . 5 ({𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
10859, 107syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) ∈ ℕ0*)
109 hashxnn0 13375 . . . . 5 ({𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}} ∈ V → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) ∈ ℕ0*)
11091, 109syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}}) ∈ ℕ0*)
111 hashxnn0 13375 . . . . 5 ({𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}} ∈ V → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ∈ ℕ0*)
11293, 111syl 17 . . . 4 (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}}) ∈ ℕ0*)
113106, 108, 110, 112xnn0add4d 12379 . . 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}}))))
114104, 113eqtrd 2831 . 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}}))))
115 vtxdun.n . . . 4 (𝜑𝑁𝑉)
116 vtxdun.vu . . . 4 (𝜑 → (Vtx‘𝑈) = 𝑉)
117115, 116eleqtrrd 2879 . . 3 (𝜑𝑁 ∈ (Vtx‘𝑈))
118 eqid 2797 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
119 eqid 2797 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
120 eqid 2797 . . . 4 dom (iEdg‘𝑈) = dom (iEdg‘𝑈)
121118, 119, 120vtxdgval 26710 . . 3 (𝑁 ∈ (Vtx‘𝑈) → ((VtxDeg‘𝑈)‘𝑁) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
122117, 121syl 17 . 2 (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
123 vtxdun.vg . . . . 5 𝑉 = (Vtx‘𝐺)
124 eqid 2797 . . . . 5 dom 𝐼 = dom 𝐼
125123, 50, 124vtxdgval 26710 . . . 4 (𝑁𝑉 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})))
126115, 125syl 17 . . 3 (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐼𝑁 ∈ (𝐼𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) = {𝑁}})))
127 vtxdun.vh . . . . 5 (𝜑 → (Vtx‘𝐻) = 𝑉)
128115, 127eleqtrrd 2879 . . . 4 (𝜑𝑁 ∈ (Vtx‘𝐻))
129 eqid 2797 . . . . 5 (Vtx‘𝐻) = (Vtx‘𝐻)
130 eqid 2797 . . . . 5 dom 𝐽 = dom 𝐽
131129, 55, 130vtxdgval 26710 . . . 4 (𝑁 ∈ (Vtx‘𝐻) → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
132128, 131syl 17 . . 3 (𝜑 → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽𝑁 ∈ (𝐽𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) = {𝑁}})))
133126, 132oveq12d 6894 . 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}}))))
134114, 122, 1333eqtr4d 2841 1 (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874   = wceq 1653  wcel 2157  {cab 2783  {crab 3091  Vcvv 3383  cun 3765  cin 3766  wss 3767  c0 4113  {csn 4366  dom cdm 5310  Fun wfun 6093   Fn wfn 6094  cfv 6099  (class class class)co 6876  0*cxnn0 11648   +𝑒 cxad 12187  chash 13366  Vtxcvtx 26223  iEdgciedg 26224  VtxDegcvtxdg 26707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-oadd 7801  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-card 9049  df-cda 9276  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-n0 11577  df-xnn0 11649  df-z 11663  df-uz 11927  df-xadd 12190  df-hash 13367  df-vtxdg 26708
This theorem is referenced by:  vtxdfiun  26724  vtxduhgrun  26725  p1evtxdeqlem  26754
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