| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-rab 3437 | . . . . . . . 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) = (𝐼 ∪ 𝐽)) | 
| 3 | 2 | dmeqd 5916 | . . . . . . . . . . . . . 14
⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐼 ∪ 𝐽)) | 
| 4 |  | dmun 5921 | . . . . . . . . . . . . . 14
⊢ dom
(𝐼 ∪ 𝐽) = (dom 𝐼 ∪ dom 𝐽) | 
| 5 | 3, 4 | eqtrdi 2793 | . . . . . . . . . . . . 13
⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽)) | 
| 6 | 5 | eleq2d 2827 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽))) | 
| 7 |  | elun 4153 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ (dom 𝐼 ∪ dom 𝐽) ↔ (𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽)) | 
| 8 | 6, 7 | bitrdi 287 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ (𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽))) | 
| 9 | 8 | anbi1d 631 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))) | 
| 10 |  | andir 1011 | . . . . . . . . . 10
⊢ (((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))) | 
| 11 | 9, 10 | bitrdi 287 | . . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))) | 
| 12 | 11 | abbidv 2808 | . . . . . . . 8
⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))}) | 
| 13 | 1, 12 | eqtrid 2789 | . . . . . . 7
⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))}) | 
| 14 |  | unab 4308 | . . . . . . . . 9
⊢ ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} | 
| 15 | 14 | eqcomi 2746 | . . . . . . . 8
⊢ {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) | 
| 16 | 15 | a1i 11 | . . . . . . 7
⊢ (𝜑 → {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))})) | 
| 17 |  | df-rab 3437 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} | 
| 18 | 2 | fveq1d 6908 | . . . . . . . . . . . . 13
⊢ (𝜑 → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥)) | 
| 19 | 18 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥)) | 
| 20 |  | vtxdun.fi | . . . . . . . . . . . . . . 15
⊢ (𝜑 → Fun 𝐼) | 
| 21 | 20 | funfnd 6597 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 Fn dom 𝐼) | 
| 22 | 21 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼) | 
| 23 |  | vtxdun.fj | . . . . . . . . . . . . . . 15
⊢ (𝜑 → Fun 𝐽) | 
| 24 | 23 | funfnd 6597 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐽 Fn dom 𝐽) | 
| 25 | 24 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → 𝐽 Fn dom 𝐽) | 
| 26 |  | vtxdun.d | . . . . . . . . . . . . . 14
⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅) | 
| 27 | 26 | anim1i 615 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) | 
| 28 |  | fvun1 7000 | . . . . . . . . . . . . 13
⊢ ((𝐼 Fn dom 𝐼 ∧ 𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐼‘𝑥)) | 
| 29 | 22, 25, 27, 28 | syl3anc 1373 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐼‘𝑥)) | 
| 30 | 19, 29 | eqtrd 2777 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼‘𝑥)) | 
| 31 | 30 | eleq2d 2827 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼‘𝑥))) | 
| 32 | 31 | rabbidva 3443 | . . . . . . . . 9
⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) | 
| 33 | 17, 32 | eqtr3id 2791 | . . . . . . . 8
⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) | 
| 34 |  | df-rab 3437 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} | 
| 35 | 18 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥)) | 
| 36 | 21 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → 𝐼 Fn dom 𝐼) | 
| 37 | 24 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → 𝐽 Fn dom 𝐽) | 
| 38 | 26 | anim1i 615 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) | 
| 39 |  | fvun2 7001 | . . . . . . . . . . . . 13
⊢ ((𝐼 Fn dom 𝐼 ∧ 𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐽‘𝑥)) | 
| 40 | 36, 37, 38, 39 | syl3anc 1373 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐽‘𝑥)) | 
| 41 | 35, 40 | eqtrd 2777 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽‘𝑥)) | 
| 42 | 41 | eleq2d 2827 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽‘𝑥))) | 
| 43 | 42 | rabbidva 3443 | . . . . . . . . 9
⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) | 
| 44 | 34, 43 | eqtr3id 2791 | . . . . . . . 8
⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) | 
| 45 | 33, 44 | uneq12d 4169 | . . . . . . 7
⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) | 
| 46 | 13, 16, 45 | 3eqtrd 2781 | . . . . . 6
⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) | 
| 47 | 46 | fveq2d 6910 | . . . . 5
⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = (♯‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))) | 
| 48 |  | vtxdun.i | . . . . . . . . . 10
⊢ 𝐼 = (iEdg‘𝐺) | 
| 49 | 48 | fvexi 6920 | . . . . . . . . 9
⊢ 𝐼 ∈ V | 
| 50 | 49 | dmex 7931 | . . . . . . . 8
⊢ dom 𝐼 ∈ V | 
| 51 | 50 | rabex 5339 | . . . . . . 7
⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V | 
| 52 | 51 | a1i 11 | . . . . . 6
⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V) | 
| 53 |  | vtxdun.j | . . . . . . . . . 10
⊢ 𝐽 = (iEdg‘𝐻) | 
| 54 | 53 | fvexi 6920 | . . . . . . . . 9
⊢ 𝐽 ∈ V | 
| 55 | 54 | dmex 7931 | . . . . . . . 8
⊢ dom 𝐽 ∈ V | 
| 56 | 55 | rabex 5339 | . . . . . . 7
⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V | 
| 57 | 56 | a1i 11 | . . . . . 6
⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V) | 
| 58 |  | ssrab2 4080 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ⊆ dom 𝐼 | 
| 59 |  | ssrab2 4080 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ⊆ dom 𝐽 | 
| 60 |  | ss2in 4245 | . . . . . . . . 9
⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)) | 
| 61 | 58, 59, 60 | mp2an 692 | . . . . . . . 8
⊢ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽) | 
| 62 | 61, 26 | sseqtrid 4026 | . . . . . . 7
⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ ∅) | 
| 63 |  | ss0 4402 | . . . . . . 7
⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅) | 
| 64 | 62, 63 | syl 17 | . . . . . 6
⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅) | 
| 65 |  | hashunx 14425 | . . . . . 6
⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V ∧ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V ∧ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅) → (♯‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))) | 
| 66 | 52, 57, 64, 65 | syl3anc 1373 | . . . . 5
⊢ (𝜑 → (♯‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))) | 
| 67 | 47, 66 | eqtrd 2777 | . . . 4
⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))) | 
| 68 |  | df-rab 3437 | . . . . . . . 8
⊢ {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} | 
| 69 | 8 | anbi1d 631 | . . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))) | 
| 70 |  | andir 1011 | . . . . . . . . . 10
⊢ (((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))) | 
| 71 | 69, 70 | bitrdi 287 | . . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))) | 
| 72 | 71 | abbidv 2808 | . . . . . . . 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}))}) | 
| 73 | 68, 72 | eqtrid 2789 | . . . . . . 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 4308 | . . . . . . . . 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}))} | 
| 75 | 74 | eqcomi 2746 | . . . . . . . 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})}) | 
| 76 | 75 | a1i 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 3437 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} | 
| 78 | 30 | eqeq1d 2739 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼‘𝑥) = {𝑁})) | 
| 79 | 78 | rabbidva 3443 | . . . . . . . . 9
⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) | 
| 80 | 77, 79 | eqtr3id 2791 | . . . . . . . 8
⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) | 
| 81 |  | df-rab 3437 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} | 
| 82 | 41 | eqeq1d 2739 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽‘𝑥) = {𝑁})) | 
| 83 | 82 | rabbidva 3443 | . . . . . . . . 9
⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) | 
| 84 | 81, 83 | eqtr3id 2791 | . . . . . . . 8
⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) | 
| 85 | 80, 84 | uneq12d 4169 | . . . . . . 7
⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}) = ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) | 
| 86 | 73, 76, 85 | 3eqtrd 2781 | . . . . . 6
⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) | 
| 87 | 86 | fveq2d 6910 | . . . . 5
⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) | 
| 88 | 50 | rabex 5339 | . . . . . . 7
⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V | 
| 89 | 88 | a1i 11 | . . . . . 6
⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V) | 
| 90 | 55 | rabex 5339 | . . . . . . 7
⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V | 
| 91 | 90 | a1i 11 | . . . . . 6
⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V) | 
| 92 |  | ssrab2 4080 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ⊆ dom 𝐼 | 
| 93 |  | ssrab2 4080 | . . . . . . . . 9
⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ⊆ dom 𝐽 | 
| 94 |  | ss2in 4245 | . . . . . . . . 9
⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)) | 
| 95 | 92, 93, 94 | mp2an 692 | . . . . . . . 8
⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽) | 
| 96 | 95, 26 | sseqtrid 4026 | . . . . . . 7
⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ ∅) | 
| 97 |  | ss0 4402 | . . . . . . 7
⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅) | 
| 98 | 96, 97 | syl 17 | . . . . . 6
⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅) | 
| 99 |  | hashunx 14425 | . . . . . 6
⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V ∧ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅) → (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) | 
| 100 | 89, 91, 98, 99 | syl3anc 1373 | . . . . 5
⊢ (𝜑 → (♯‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) | 
| 101 | 87, 100 | eqtrd 2777 | . . . 4
⊢ (𝜑 → (♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) | 
| 102 | 67, 101 | oveq12d 7449 | . . 3
⊢ (𝜑 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom (iEdg‘𝑈) ∣
((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) +𝑒
((♯‘{𝑥 ∈
dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))) | 
| 103 |  | hashxnn0 14378 | . . . . 5
⊢ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈
ℕ0*) | 
| 104 | 52, 103 | syl 17 | . . . 4
⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈
ℕ0*) | 
| 105 |  | hashxnn0 14378 | . . . . 5
⊢ ({𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V → (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈
ℕ0*) | 
| 106 | 57, 105 | syl 17 | . . . 4
⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈
ℕ0*) | 
| 107 |  | hashxnn0 14378 | . . . . 5
⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈
ℕ0*) | 
| 108 | 89, 107 | syl 17 | . . . 4
⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈
ℕ0*) | 
| 109 |  | hashxnn0 14378 | . . . . 5
⊢ ({𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈
ℕ0*) | 
| 110 | 91, 109 | syl 17 | . . . 4
⊢ (𝜑 → (♯‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈
ℕ0*) | 
| 111 | 104, 106,
108, 110 | xnn0add4d 13346 | . . 3
⊢ (𝜑 → (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) +𝑒
((♯‘{𝑥 ∈
dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) +𝑒
((♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))) | 
| 112 | 102, 111 | eqtrd 2777 | . 2
⊢ (𝜑 → ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom (iEdg‘𝑈) ∣
((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) +𝑒
((♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))) | 
| 113 |  | vtxdun.n | . . . 4
⊢ (𝜑 → 𝑁 ∈ 𝑉) | 
| 114 |  | vtxdun.vu | . . . 4
⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | 
| 115 | 113, 114 | eleqtrrd 2844 | . . 3
⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝑈)) | 
| 116 |  | eqid 2737 | . . . 4
⊢
(Vtx‘𝑈) =
(Vtx‘𝑈) | 
| 117 |  | eqid 2737 | . . . 4
⊢
(iEdg‘𝑈) =
(iEdg‘𝑈) | 
| 118 |  | eqid 2737 | . . . 4
⊢ dom
(iEdg‘𝑈) = dom
(iEdg‘𝑈) | 
| 119 | 116, 117,
118 | vtxdgval 29486 | . . 3
⊢ (𝑁 ∈ (Vtx‘𝑈) → ((VtxDeg‘𝑈)‘𝑁) = ((♯‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom (iEdg‘𝑈) ∣
((iEdg‘𝑈)‘𝑥) = {𝑁}}))) | 
| 120 | 115, 119 | syl 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 2737 | . . . . 5
⊢ dom 𝐼 = dom 𝐼 | 
| 123 | 121, 48, 122 | vtxdgval 29486 | . . . 4
⊢ (𝑁 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}))) | 
| 124 | 113, 123 | syl 17 | . . 3
⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}))) | 
| 125 |  | vtxdun.vh | . . . . 5
⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | 
| 126 | 113, 125 | eleqtrrd 2844 | . . . 4
⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐻)) | 
| 127 |  | eqid 2737 | . . . . 5
⊢
(Vtx‘𝐻) =
(Vtx‘𝐻) | 
| 128 |  | eqid 2737 | . . . . 5
⊢ dom 𝐽 = dom 𝐽 | 
| 129 | 127, 53, 128 | vtxdgval 29486 | . . . 4
⊢ (𝑁 ∈ (Vtx‘𝐻) → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) | 
| 130 | 126, 129 | syl 17 | . . 3
⊢ (𝜑 → ((VtxDeg‘𝐻)‘𝑁) = ((♯‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) | 
| 131 | 124, 130 | oveq12d 7449 | . 2
⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)) = (((♯‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) +𝑒
((♯‘{𝑥 ∈
dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +𝑒
(♯‘{𝑥 ∈
dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))) | 
| 132 | 112, 120,
131 | 3eqtr4d 2787 | 1
⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁))) |