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Mirrors > Home > MPE Home > Th. List > umgredgprv | Structured version Visualization version GIF version |
Description: In a multigraph, an edge is an unordered pair of vertices. This theorem would not hold for arbitrary hyper-/pseudographs since either 𝑀 or 𝑁 could be proper classes ((𝐸‘𝑋) would be a loop in this case), which are no vertices of course. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.) |
Ref | Expression |
---|---|
umgrnloopv.e | ⊢ 𝐸 = (iEdg‘𝐺) |
umgredgprv.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
umgredgprv | ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgruhgr 26891 | . . 3 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | |
2 | umgredgprv.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | umgrnloopv.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | 2, 3 | uhgrss 26851 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ⊆ 𝑉) |
5 | 1, 4 | sylan 582 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ⊆ 𝑉) |
6 | 2, 3 | umgredg2 26887 | . 2 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸‘𝑋)) = 2) |
7 | sseq1 3994 | . . . . 5 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((𝐸‘𝑋) ⊆ 𝑉 ↔ {𝑀, 𝑁} ⊆ 𝑉)) | |
8 | fveqeq2 6681 | . . . . 5 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → ((♯‘(𝐸‘𝑋)) = 2 ↔ (♯‘{𝑀, 𝑁}) = 2)) | |
9 | 7, 8 | anbi12d 632 | . . . 4 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → (((𝐸‘𝑋) ⊆ 𝑉 ∧ (♯‘(𝐸‘𝑋)) = 2) ↔ ({𝑀, 𝑁} ⊆ 𝑉 ∧ (♯‘{𝑀, 𝑁}) = 2))) |
10 | eqid 2823 | . . . . . . 7 ⊢ {𝑀, 𝑁} = {𝑀, 𝑁} | |
11 | 10 | hashprdifel 13762 | . . . . . 6 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁} ∧ 𝑀 ≠ 𝑁)) |
12 | prssg 4754 | . . . . . . . 8 ⊢ ((𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁}) → ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ {𝑀, 𝑁} ⊆ 𝑉)) | |
13 | 12 | 3adant3 1128 | . . . . . . 7 ⊢ ((𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁} ∧ 𝑀 ≠ 𝑁) → ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ {𝑀, 𝑁} ⊆ 𝑉)) |
14 | 13 | biimprd 250 | . . . . . 6 ⊢ ((𝑀 ∈ {𝑀, 𝑁} ∧ 𝑁 ∈ {𝑀, 𝑁} ∧ 𝑀 ≠ 𝑁) → ({𝑀, 𝑁} ⊆ 𝑉 → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
15 | 11, 14 | syl 17 | . . . . 5 ⊢ ((♯‘{𝑀, 𝑁}) = 2 → ({𝑀, 𝑁} ⊆ 𝑉 → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
16 | 15 | impcom 410 | . . . 4 ⊢ (({𝑀, 𝑁} ⊆ 𝑉 ∧ (♯‘{𝑀, 𝑁}) = 2) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) |
17 | 9, 16 | syl6bi 255 | . . 3 ⊢ ((𝐸‘𝑋) = {𝑀, 𝑁} → (((𝐸‘𝑋) ⊆ 𝑉 ∧ (♯‘(𝐸‘𝑋)) = 2) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
18 | 17 | com12 32 | . 2 ⊢ (((𝐸‘𝑋) ⊆ 𝑉 ∧ (♯‘(𝐸‘𝑋)) = 2) → ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
19 | 5, 6, 18 | syl2anc 586 | 1 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 {cpr 4571 dom cdm 5557 ‘cfv 6357 2c2 11695 ♯chash 13693 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 UMGraphcumgr 26868 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 df-uhgr 26845 df-upgr 26869 df-umgr 26870 |
This theorem is referenced by: umgrnloop 26895 usgredgprv 26978 |
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