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Mirrors > Home > MPE Home > Th. List > umgrun | Structured version Visualization version GIF version |
Description: The union 𝑈 of two multigraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a multigraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
umgrun.g | ⊢ (𝜑 → 𝐺 ∈ UMGraph) |
umgrun.h | ⊢ (𝜑 → 𝐻 ∈ UMGraph) |
umgrun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
umgrun.f | ⊢ 𝐹 = (iEdg‘𝐻) |
umgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
umgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
umgrun.i | ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) |
umgrun.u | ⊢ (𝜑 → 𝑈 ∈ 𝑊) |
umgrun.v | ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) |
umgrun.un | ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) |
Ref | Expression |
---|---|
umgrun | ⊢ (𝜑 → 𝑈 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgrun.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ UMGraph) | |
2 | umgrun.vg | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | umgrun.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | 2, 3 | umgrf 26883 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
6 | umgrun.h | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ UMGraph) | |
7 | eqid 2821 | . . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻) | |
8 | umgrun.f | . . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻) | |
9 | 7, 8 | umgrf 26883 | . . . . . 6 ⊢ (𝐻 ∈ UMGraph → 𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) |
10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) |
11 | umgrun.vh | . . . . . . . . 9 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
12 | 11 | eqcomd 2827 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻)) |
13 | 12 | pweqd 4558 | . . . . . . 7 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻)) |
14 | 13 | rabeqdv 3484 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) |
15 | 14 | feq3d 6501 | . . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2})) |
16 | 10, 15 | mpbird 259 | . . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
17 | umgrun.i | . . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
18 | 5, 16, 17 | fun2d 6542 | . . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
19 | umgrun.un | . . . 4 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) | |
20 | 19 | dmeqd 5774 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹)) |
21 | dmun 5779 | . . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹) | |
22 | 20, 21 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹)) |
23 | umgrun.v | . . . . . 6 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
24 | 23 | pweqd 4558 | . . . . 5 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉) |
25 | 24 | rabeqdv 3484 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘𝑈) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
26 | 19, 22, 25 | feq123d 6503 | . . 3 ⊢ (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ 𝒫 (Vtx‘𝑈) ∣ (♯‘𝑥) = 2} ↔ (𝐸 ∪ 𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) |
27 | 18, 26 | mpbird 259 | . 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ 𝒫 (Vtx‘𝑈) ∣ (♯‘𝑥) = 2}) |
28 | umgrun.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊) | |
29 | eqid 2821 | . . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈) | |
30 | eqid 2821 | . . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈) | |
31 | 29, 30 | isumgrs 26881 | . . 3 ⊢ (𝑈 ∈ 𝑊 → (𝑈 ∈ UMGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ 𝒫 (Vtx‘𝑈) ∣ (♯‘𝑥) = 2})) |
32 | 28, 31 | syl 17 | . 2 ⊢ (𝜑 → (𝑈 ∈ UMGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ 𝒫 (Vtx‘𝑈) ∣ (♯‘𝑥) = 2})) |
33 | 27, 32 | mpbird 259 | 1 ⊢ (𝜑 → 𝑈 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {crab 3142 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 𝒫 cpw 4539 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 2c2 11693 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 UMGraphcumgr 26866 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-umgr 26868 |
This theorem is referenced by: umgrunop 26906 usgrun 26972 |
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