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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 26719 | . 2 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
7 | usgr1e.e | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
8 | hashprg 13560 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 ≠ 𝐶 ↔ (♯‘{𝐵, 𝐶}) = 2)) | |
9 | 3, 4, 8 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → (𝐵 ≠ 𝐶 ↔ (♯‘{𝐵, 𝐶}) = 2)) |
10 | 7, 9 | mpbid 224 | . . . 4 ⊢ (𝜑 → (♯‘{𝐵, 𝐶}) = 2) |
11 | prex 5183 | . . . . 5 ⊢ {𝐵, 𝐶} ∈ V | |
12 | fveqeq2 6502 | . . . . 5 ⊢ (𝑥 = {𝐵, 𝐶} → ((♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2)) | |
13 | 11, 12 | ralsn 4487 | . . . 4 ⊢ (∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2 ↔ (♯‘{𝐵, 𝐶}) = 2) |
14 | 10, 13 | sylibr 226 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2) |
15 | edgval 26527 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
17 | 5 | rneqd 5644 | . . . . 5 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {〈𝐴, {𝐵, 𝐶}〉}) |
18 | rnsnopg 5911 | . . . . . 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 2812 | . . . 4 ⊢ (𝜑 → (Edg‘𝐺) = {{𝐵, 𝐶}}) |
21 | 20 | raleqdv 3349 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2 ↔ ∀𝑥 ∈ {{𝐵, 𝐶}} (♯‘𝑥) = 2)) |
22 | 14, 21 | mpbird 249 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2) |
23 | usgruspgrb 26659 | . 2 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) = 2)) | |
24 | 6, 22, 23 | sylanbrc 575 | 1 ⊢ (𝜑 → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ∀wral 3082 {csn 4435 {cpr 4437 〈cop 4441 ran crn 5401 ‘cfv 6182 2c2 11488 ♯chash 13498 Vtxcvtx 26474 iEdgciedg 26475 Edgcedg 26525 USPGraphcuspgr 26626 USGraphcusgr 26627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-n0 11701 df-xnn0 11773 df-z 11787 df-uz 12052 df-fz 12702 df-hash 13499 df-edg 26526 df-uspgr 26628 df-usgr 26629 |
This theorem is referenced by: usgr1eop 26725 1egrvtxdg1 26984 1egrvtxdg0 26986 |
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