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Mirrors > Home > MPE Home > Th. List > uspgr1e | Structured version Visualization version GIF version |
Description: A simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.) |
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}〉}) |
Ref | Expression |
---|---|
uspgr1e | ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgr1e.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | prex 5298 | . . . . . . 7 ⊢ {𝐵, 𝐶} ∈ V | |
3 | 2 | snid 4561 | . . . . . 6 ⊢ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}} |
4 | f1sng 6631 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}}) → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{{𝐵, 𝐶}}) | |
5 | 1, 3, 4 | sylancl 589 | . . . . 5 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{{𝐵, 𝐶}}) |
6 | uspgr1e.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
7 | uspgr1e.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
8 | 6, 7 | prssd 4715 | . . . . . . . 8 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝑉) |
9 | uspgr1e.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 8, 9 | sseqtrdi 3965 | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
11 | 2 | elpw 4501 | . . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺) ↔ {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
12 | 10, 11 | sylibr 237 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺)) |
13 | 12, 6 | upgr1elem 26905 | . . . . 5 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
14 | f1ss 6555 | . . . . 5 ⊢ (({〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{{𝐵, 𝐶}} ∧ {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
15 | 5, 13, 14 | syl2anc 587 | . . . 4 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
16 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ∈ V) |
17 | 16, 6 | upgr1elem 26905 | . . . . . 6 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
18 | f1ss 6555 | . . . . . 6 ⊢ (({〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{{𝐵, 𝐶}} ∧ {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
19 | 5, 17, 18 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
20 | f1dm 6553 | . . . . 5 ⊢ ({〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → dom {〈𝐴, {𝐵, 𝐶}〉} = {𝐴}) | |
21 | f1eq2 6545 | . . . . 5 ⊢ (dom {〈𝐴, {𝐵, 𝐶}〉} = {𝐴} → ({〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) | |
22 | 19, 20, 21 | 3syl 18 | . . . 4 ⊢ (𝜑 → ({〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
23 | 15, 22 | mpbird 260 | . . 3 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
24 | uspgr1e.e | . . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) | |
25 | 24 | dmeqd 5738 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝐵, 𝐶}〉}) |
26 | eqidd 2799 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
27 | 24, 25, 26 | f1eq123d 6583 | . . 3 ⊢ (𝜑 → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ {〈𝐴, {𝐵, 𝐶}〉}:dom {〈𝐴, {𝐵, 𝐶}〉}–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
28 | 23, 27 | mpbird 260 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
29 | 9 | 1vgrex 26795 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐺 ∈ V) |
30 | eqid 2798 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
31 | eqid 2798 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
32 | 30, 31 | isuspgr 26945 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
33 | 6, 29, 32 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
34 | 28, 33 | mpbird 260 | 1 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 {cpr 4527 〈cop 4531 class class class wbr 5030 dom cdm 5519 –1-1→wf1 6321 ‘cfv 6324 ≤ cle 10665 2c2 11680 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 USPGraphcuspgr 26941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-uspgr 26943 |
This theorem is referenced by: usgr1e 27035 uspgr1eop 27037 1loopgruspgr 27290 |
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