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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 28172 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UMGraph) |
4 | usgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USGraph) | |
5 | usgrumgr 28172 | . . 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 28113 | 1 ⊢ (𝜑 → 𝑈 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 dom cdm 5634 ‘cfv 6497 Vtxcvtx 27989 iEdgciedg 27990 UMGraphcumgr 28074 USGraphcusgr 28142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 df-umgr 28076 df-usgr 28144 |
This theorem is referenced by: (None) |
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