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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 5326 | . . . . . . 7 ⊢ {𝐵, 𝐶} ∈ V | |
3 | 2 | snid 4594 | . . . . . 6 ⊢ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}} |
4 | f1sng 6649 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝐵, 𝐶} ∈ {{𝐵, 𝐶}}) → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{{𝐵, 𝐶}}) | |
5 | 1, 3, 4 | sylancl 588 | . . . . 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 4748 | . . . . . . . 8 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝑉) |
9 | uspgr1e.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 8, 9 | sseqtrdi 4010 | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
11 | 2 | elpw 4536 | . . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺) ↔ {𝐵, 𝐶} ⊆ (Vtx‘𝐺)) |
12 | 10, 11 | sylibr 236 | . . . . . 6 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝒫 (Vtx‘𝐺)) |
13 | 12, 6 | upgr1elem 26893 | . . . . 5 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
14 | f1ss 6573 | . . . . 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 586 | . . . 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 26893 | . . . . . 6 ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
18 | f1ss 6573 | . . . . . 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 586 | . . . . 5 ⊢ (𝜑 → {〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
20 | f1dm 6572 | . . . . 5 ⊢ ({〈𝐴, {𝐵, 𝐶}〉}:{𝐴}–1-1→{𝑥 ∈ (V ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → dom {〈𝐴, {𝐵, 𝐶}〉} = {𝐴}) | |
21 | f1eq2 6564 | . . . . 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 259 | . . 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 5767 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, {𝐵, 𝐶}〉}) |
26 | eqidd 2821 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
27 | 24, 25, 26 | f1eq123d 6601 | . . 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 259 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
29 | 9 | 1vgrex 26783 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐺 ∈ V) |
30 | eqid 2820 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
31 | eqid 2820 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
32 | 30, 31 | isuspgr 26933 | . . 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 259 | 1 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 {crab 3141 Vcvv 3491 ∖ cdif 3926 ⊆ wss 3929 ∅c0 4284 𝒫 cpw 4532 {csn 4560 {cpr 4562 〈cop 4566 class class class wbr 5059 dom cdm 5548 –1-1→wf1 6345 ‘cfv 6348 ≤ cle 10669 2c2 11686 ♯chash 13687 Vtxcvtx 26777 iEdgciedg 26778 USPGraphcuspgr 26929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12890 df-hash 13688 df-uspgr 26931 |
This theorem is referenced by: usgr1e 27023 uspgr1eop 27025 1loopgruspgr 27278 |
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