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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 27592 | . 2 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
7 | usgr1e.e | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
8 | hashprg 14091 | . . . . . 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 5358 | . . . . 5 ⊢ {𝐵, 𝐶} ∈ V | |
12 | fveqeq2 6777 | . . . . 5 ⊢ (𝑥 = {𝐵, 𝐶} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2)) | |
13 | 11, 12 | ralsn 4622 | . . . 4 ⊢ (∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2) |
14 | 10, 13 | sylibr 233 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2) |
15 | edgval 27400 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
17 | 5 | rneqd 5844 | . . . . 5 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {〈𝐴, {𝐵, 𝐶}〉}) |
18 | rnsnopg 6121 | . . . . . 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 2783 | . . . 4 ⊢ (𝜑 → (Edg‘𝐺) = {{𝐵, 𝐶}}) |
21 | 20 | raleqdv 3346 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2 ↔ ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2)) |
22 | 14, 21 | mpbird 256 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2) |
23 | usgruspgrb 27532 | . 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 1541 ∈ wcel 2109 ≠ wne 2944 ∀wral 3065 {csn 4566 {cpr 4568 〈cop 4572 ran crn 5589 ‘cfv 6430 2c2 12011 ♯chash 14025 Vtxcvtx 27347 iEdgciedg 27348 Edgcedg 27398 USPGraphcuspgr 27499 USGraphcusgr 27500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-oadd 8285 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-n0 12217 df-xnn0 12289 df-z 12303 df-uz 12565 df-fz 13222 df-hash 14026 df-edg 27399 df-uspgr 27501 df-usgr 27502 |
This theorem is referenced by: usgr1eop 27598 1egrvtxdg1 27857 1egrvtxdg0 27859 |
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