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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 29130 | . . . . 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 2735 | . . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻) | |
8 | umgrun.f | . . . . . . 7 ⊢ 𝐹 = (iEdg‘𝐻) | |
9 | 7, 8 | umgrf 29130 | . . . . . 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 2741 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘𝐻)) |
13 | 12 | pweqd 4622 | . . . . . . 7 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻)) |
14 | 13 | rabeqdv 3449 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) |
15 | 14 | feq3d 6724 | . . . . 5 ⊢ (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2})) |
16 | 10, 15 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐹:dom 𝐹⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
17 | umgrun.i | . . . 4 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
18 | 5, 16, 17 | fun2d 6773 | . . 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 5919 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐸 ∪ 𝐹)) |
21 | dmun 5924 | . . . . 5 ⊢ dom (𝐸 ∪ 𝐹) = (dom 𝐸 ∪ dom 𝐹) | |
22 | 20, 21 | eqtrdi 2791 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹)) |
23 | umgrun.v | . . . . . 6 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
24 | 23 | pweqd 4622 | . . . . 5 ⊢ (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉) |
25 | 24 | rabeqdv 3449 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘𝑈) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
26 | 19, 22, 25 | feq123d 6726 | . . 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 257 | . 2 ⊢ (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ 𝒫 (Vtx‘𝑈) ∣ (♯‘𝑥) = 2}) |
28 | umgrun.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊) | |
29 | eqid 2735 | . . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈) | |
30 | eqid 2735 | . . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈) | |
31 | 29, 30 | isumgrs 29128 | . . 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 257 | 1 ⊢ (𝜑 → 𝑈 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 {crab 3433 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 𝒫 cpw 4605 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 2c2 12319 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 UMGraphcumgr 29113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-umgr 29115 |
This theorem is referenced by: umgrunop 29153 usgrun 29222 |
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