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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 29005 | . 2 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
7 | usgr1e.e | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
8 | hashprg 14358 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 ≠ 𝐶 ↔ (♯‘{𝐵, 𝐶}) = 2)) | |
9 | 3, 4, 8 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝐵 ≠ 𝐶 ↔ (♯‘{𝐵, 𝐶}) = 2)) |
10 | 7, 9 | mpbid 231 | . . . 4 ⊢ (𝜑 → (♯‘{𝐵, 𝐶}) = 2) |
11 | prex 5425 | . . . . 5 ⊢ {𝐵, 𝐶} ∈ V | |
12 | fveqeq2 6893 | . . . . 5 ⊢ (𝑥 = {𝐵, 𝐶} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2)) | |
13 | 11, 12 | ralsn 4680 | . . . 4 ⊢ (∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2) |
14 | 10, 13 | sylibr 233 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2) |
15 | edgval 28813 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
17 | 5 | rneqd 5930 | . . . . 5 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝐵, 𝐶}⟩}) |
18 | rnsnopg 6213 | . . . . . 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 2770 | . . . 4 ⊢ (𝜑 → (Edg‘𝐺) = {{𝐵, 𝐶}}) |
21 | 20 | raleqdv 3319 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2 ↔ ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2)) |
22 | 14, 21 | mpbird 257 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2) |
23 | usgruspgrb 28945 | . 2 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2)) | |
24 | 6, 22, 23 | sylanbrc 582 | 1 ⊢ (𝜑 → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 {csn 4623 {cpr 4625 ⟨cop 4629 ran crn 5670 ‘cfv 6536 2c2 12268 ♯chash 14293 Vtxcvtx 28760 iEdgciedg 28761 Edgcedg 28811 USPGraphcuspgr 28912 USGraphcusgr 28913 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-hash 14294 df-edg 28812 df-uspgr 28914 df-usgr 28915 |
This theorem is referenced by: usgr1eop 29011 1egrvtxdg1 29271 1egrvtxdg0 29273 |
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