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Mirrors > Home > MPE Home > Th. List > usgrun | Structured version Visualization version GIF version |
Description: The union 𝑈 of two simple graphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a multigraph (not necessarily a simple graph!) with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) |
Ref | Expression |
---|---|
usgrun.g | ⊢ (𝜑 → 𝐺 ∈ USGraph) |
usgrun.h | ⊢ (𝜑 → 𝐻 ∈ USGraph) |
usgrun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
usgrun.f | ⊢ 𝐹 = (iEdg‘𝐻) |
usgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
usgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
usgrun.i | ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) |
usgrun.u | ⊢ (𝜑 → 𝑈 ∈ 𝑊) |
usgrun.v | ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) |
usgrun.un | ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) |
Ref | Expression |
---|---|
usgrun | ⊢ (𝜑 → 𝑈 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ USGraph) | |
2 | usgrumgr 27124 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UMGraph) |
4 | usgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USGraph) | |
5 | usgrumgr 27124 | . . 3 ⊢ (𝐻 ∈ USGraph → 𝐻 ∈ UMGraph) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐻 ∈ UMGraph) |
7 | usgrun.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
8 | usgrun.f | . 2 ⊢ 𝐹 = (iEdg‘𝐻) | |
9 | usgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | usgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
11 | usgrun.i | . 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
12 | usgrun.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑊) | |
13 | usgrun.v | . 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
14 | usgrun.un | . 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) | |
15 | 3, 6, 7, 8, 9, 10, 11, 12, 13, 14 | umgrun 27065 | 1 ⊢ (𝜑 → 𝑈 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∪ cun 3841 ∩ cin 3842 ∅c0 4211 dom cdm 5525 ‘cfv 6339 Vtxcvtx 26941 iEdgciedg 26942 UMGraphcumgr 27026 USGraphcusgr 27094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-hash 13783 df-umgr 27028 df-usgr 27096 |
This theorem is referenced by: (None) |
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