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Mirrors > Home > MPE Home > Th. List > usgredg3 | Structured version Visualization version GIF version |
Description: The value of the "edge function" of a simple graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.) |
Ref | Expression |
---|---|
usgredg3.v | ⊢ 𝑉 = (Vtx‘𝐺) |
usgredg3.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
usgredg3 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ (𝐸‘𝑋) = {𝑥, 𝑦})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrfun 28999 | . . . . 5 ⊢ (𝐺 ∈ USGraph → Fun (iEdg‘𝐺)) | |
2 | usgredg3.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 2 | funeqi 6579 | . . . . 5 ⊢ (Fun 𝐸 ↔ Fun (iEdg‘𝐺)) |
4 | 1, 3 | sylibr 233 | . . . 4 ⊢ (𝐺 ∈ USGraph → Fun 𝐸) |
5 | fvelrn 7091 | . . . 4 ⊢ ((Fun 𝐸 ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ ran 𝐸) | |
6 | 4, 5 | sylan 578 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ ran 𝐸) |
7 | edgval 28890 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
9 | 2 | eqcomi 2737 | . . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸 |
10 | 9 | rneqi 5943 | . . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐸 |
11 | 8, 10 | eqtrdi 2784 | . . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐸) |
12 | 11 | adantr 479 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (Edg‘𝐺) = ran 𝐸) |
13 | 6, 12 | eleqtrrd 2832 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ (Edg‘𝐺)) |
14 | usgredg3.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
15 | eqid 2728 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
16 | 14, 15 | usgredg 29040 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (𝐸‘𝑋) ∈ (Edg‘𝐺)) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ (𝐸‘𝑋) = {𝑥, 𝑦})) |
17 | 13, 16 | syldan 589 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ (𝐸‘𝑋) = {𝑥, 𝑦})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 {cpr 4634 dom cdm 5682 ran crn 5683 Fun wfun 6547 ‘cfv 6553 Vtxcvtx 28837 iEdgciedg 28838 Edgcedg 28888 USGraphcusgr 28990 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-dju 9934 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-hash 14332 df-edg 28889 df-umgr 28924 df-usgr 28992 |
This theorem is referenced by: usgredg4 29058 |
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