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Mirrors > Home > MPE Home > Th. List > usgr1e | Structured version Visualization version GIF version |
Description: A simple graph with one edge (with additional assumption that 𝐵 ≠ 𝐶 since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.) |
Ref | Expression |
---|---|
uspgr1e.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uspgr1e.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
uspgr1e.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
uspgr1e.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
uspgr1e.e | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) |
usgr1e.e | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
usgr1e | ⊢ (𝜑 → 𝐺 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgr1e.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uspgr1e.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
3 | uspgr1e.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | uspgr1e.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
5 | uspgr1e.e | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) | |
6 | 1, 2, 3, 4, 5 | uspgr1e 26953 | . 2 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
7 | usgr1e.e | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
8 | hashprg 13744 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 ≠ 𝐶 ↔ (♯‘{𝐵, 𝐶}) = 2)) | |
9 | 3, 4, 8 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐵 ≠ 𝐶 ↔ (♯‘{𝐵, 𝐶}) = 2)) |
10 | 7, 9 | mpbid 233 | . . . 4 ⊢ (𝜑 → (♯‘{𝐵, 𝐶}) = 2) |
11 | prex 5323 | . . . . 5 ⊢ {𝐵, 𝐶} ∈ V | |
12 | fveqeq2 6672 | . . . . 5 ⊢ (𝑥 = {𝐵, 𝐶} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2)) | |
13 | 11, 12 | ralsn 4611 | . . . 4 ⊢ (∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2) |
14 | 10, 13 | sylibr 235 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2) |
15 | edgval 26761 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
17 | 5 | rneqd 5801 | . . . . 5 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {〈𝐴, {𝐵, 𝐶}〉}) |
18 | rnsnopg 6071 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → ran {〈𝐴, {𝐵, 𝐶}〉} = {{𝐵, 𝐶}}) | |
19 | 2, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → ran {〈𝐴, {𝐵, 𝐶}〉} = {{𝐵, 𝐶}}) |
20 | 16, 17, 19 | 3eqtrd 2857 | . . . 4 ⊢ (𝜑 → (Edg‘𝐺) = {{𝐵, 𝐶}}) |
21 | 20 | raleqdv 3413 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2 ↔ ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2)) |
22 | 14, 21 | mpbird 258 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2) |
23 | usgruspgrb 26893 | . 2 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2)) | |
24 | 6, 22, 23 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 {csn 4557 {cpr 4559 〈cop 4563 ran crn 5549 ‘cfv 6348 2c2 11680 ♯chash 13678 Vtxcvtx 26708 iEdgciedg 26709 Edgcedg 26759 USPGraphcuspgr 26860 USGraphcusgr 26861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 df-edg 26760 df-uspgr 26862 df-usgr 26863 |
This theorem is referenced by: usgr1eop 26959 1egrvtxdg1 27218 1egrvtxdg0 27220 |
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