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Mirrors > Home > MPE Home > Th. List > umgrunop | Structured version Visualization version GIF version |
Description: The union of two multigraphs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are multigraphs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a multigraph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised 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) = ∅) |
Ref | Expression |
---|---|
umgrunop | ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgrun.g | . 2 ⊢ (𝜑 → 𝐺 ∈ UMGraph) | |
2 | umgrun.h | . 2 ⊢ (𝜑 → 𝐻 ∈ UMGraph) | |
3 | umgrun.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | umgrun.f | . 2 ⊢ 𝐹 = (iEdg‘𝐻) | |
5 | umgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | umgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
7 | umgrun.i | . 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
8 | opex 5454 | . . 3 ⊢ 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V) |
10 | 5 | fvexi 6895 | . . . 4 ⊢ 𝑉 ∈ V |
11 | 3 | fvexi 6895 | . . . . 5 ⊢ 𝐸 ∈ V |
12 | 4 | fvexi 6895 | . . . . 5 ⊢ 𝐹 ∈ V |
13 | 11, 12 | unex 7726 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
14 | 10, 13 | pm3.2i 470 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
15 | opvtxfv 28688 | . . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (Vtx‘〈𝑉, (𝐸 ∪ 𝐹)〉) = 𝑉) | |
16 | 14, 15 | mp1i 13 | . 2 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐸 ∪ 𝐹)〉) = 𝑉) |
17 | opiedgfv 28691 | . . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (iEdg‘〈𝑉, (𝐸 ∪ 𝐹)〉) = (𝐸 ∪ 𝐹)) | |
18 | 14, 17 | mp1i 13 | . 2 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐸 ∪ 𝐹)〉) = (𝐸 ∪ 𝐹)) |
19 | 1, 2, 3, 4, 5, 6, 7, 9, 16, 18 | umgrun 28804 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∪ cun 3938 ∩ cin 3939 ∅c0 4314 〈cop 4626 dom cdm 5666 ‘cfv 6533 Vtxcvtx 28680 iEdgciedg 28681 UMGraphcumgr 28765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287 df-vtx 28682 df-iedg 28683 df-umgr 28767 |
This theorem is referenced by: usgrunop 28872 |
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