Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 1loopgruspgr | Structured version Visualization version GIF version |
Description: A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 27023). (Contributed by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
1loopgruspgr.v | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
1loopgruspgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1loopgruspgr.n | ⊢ (𝜑 → 𝑁 ∈ 𝑉) |
1loopgruspgr.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) |
Ref | Expression |
---|---|
1loopgruspgr | ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2819 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1loopgruspgr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
3 | 1loopgruspgr.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑉) | |
4 | 1loopgruspgr.v | . . 3 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
5 | 3, 4 | eleqtrrd 2914 | . 2 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) |
6 | 1loopgruspgr.i | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) | |
7 | dfsn2 4572 | . . . . . 6 ⊢ {𝑁} = {𝑁, 𝑁} | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → {𝑁} = {𝑁, 𝑁}) |
9 | 8 | opeq2d 4802 | . . . 4 ⊢ (𝜑 → 〈𝐴, {𝑁}〉 = 〈𝐴, {𝑁, 𝑁}〉) |
10 | 9 | sneqd 4571 | . . 3 ⊢ (𝜑 → {〈𝐴, {𝑁}〉} = {〈𝐴, {𝑁, 𝑁}〉}) |
11 | 6, 10 | eqtrd 2854 | . 2 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁, 𝑁}〉}) |
12 | 1, 2, 5, 5, 11 | uspgr1e 27018 | 1 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 {csn 4559 {cpr 4561 〈cop 4565 ‘cfv 6348 Vtxcvtx 26773 iEdgciedg 26774 USPGraphcuspgr 26925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-xnn0 11960 df-z 11974 df-uz 12236 df-fz 12885 df-hash 13683 df-uspgr 26927 |
This theorem is referenced by: 1loopgrnb0 27276 1loopgrvd2 27277 |
Copyright terms: Public domain | W3C validator |