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Mirrors > Home > MPE Home > Th. List > umgrislfupgr | Structured version Visualization version GIF version |
Description: A multigraph is a loop-free pseudograph. (Contributed by AV, 27-Jan-2021.) |
Ref | Expression |
---|---|
umgrislfupgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
umgrislfupgr.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
umgrislfupgr | ⊢ (𝐺 ∈ UMGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgrupgr 27226 | . . 3 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | |
2 | umgrislfupgr.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | umgrislfupgr.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 2, 3 | umgrf 27221 | . . . 4 ⊢ (𝐺 ∈ UMGraph → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
5 | id 22 | . . . . 5 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | |
6 | 2re 11934 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
7 | 6 | leidi 11396 | . . . . . . . . . 10 ⊢ 2 ≤ 2 |
8 | 7 | a1i 11 | . . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → 2 ≤ 2) |
9 | breq2 5074 | . . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ 2)) | |
10 | 8, 9 | mpbird 260 | . . . . . . . 8 ⊢ ((♯‘𝑥) = 2 → 2 ≤ (♯‘𝑥)) |
11 | 10 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑉 → ((♯‘𝑥) = 2 → 2 ≤ (♯‘𝑥))) |
12 | 11 | ss2rabi 4007 | . . . . . 6 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
14 | 5, 13 | fssd 6585 | . . . 4 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
15 | 4, 14 | syl 17 | . . 3 ⊢ (𝐺 ∈ UMGraph → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
16 | 1, 15 | jca 515 | . 2 ⊢ (𝐺 ∈ UMGraph → (𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})) |
17 | 2, 3 | upgrf 27209 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
18 | fin 6621 | . . . . 5 ⊢ (𝐼:dom 𝐼⟶({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ↔ (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})) | |
19 | umgrislfupgrlem 27245 | . . . . . 6 ⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} | |
20 | feq3 6550 | . . . . . 6 ⊢ (({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} → (𝐼:dom 𝐼⟶({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ (𝐼:dom 𝐼⟶({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
22 | 18, 21 | sylbb1 240 | . . . 4 ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
23 | 17, 22 | sylan 583 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
24 | 2, 3 | isumgr 27218 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UMGraph ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
25 | 24 | adantr 484 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (𝐺 ∈ UMGraph ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
26 | 23, 25 | mpbird 260 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐺 ∈ UMGraph) |
27 | 16, 26 | impbii 212 | 1 ⊢ (𝐺 ∈ UMGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {crab 3068 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4254 𝒫 cpw 4530 {csn 4558 class class class wbr 5070 dom cdm 5569 ⟶wf 6397 ‘cfv 6401 ≤ cle 10898 2c2 11915 ♯chash 13929 Vtxcvtx 27119 iEdgciedg 27120 UPGraphcupgr 27203 UMGraphcumgr 27204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 ax-un 7545 ax-cnex 10815 ax-resscn 10816 ax-1cn 10817 ax-icn 10818 ax-addcl 10819 ax-addrcl 10820 ax-mulcl 10821 ax-mulrcl 10822 ax-mulcom 10823 ax-addass 10824 ax-mulass 10825 ax-distr 10826 ax-i2m1 10827 ax-1ne0 10828 ax-1rid 10829 ax-rnegex 10830 ax-rrecex 10831 ax-cnre 10832 ax-pre-lttri 10833 ax-pre-lttrn 10834 ax-pre-ltadd 10835 ax-pre-mulgt0 10836 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5153 df-tr 5179 df-id 5472 df-eprel 5478 df-po 5486 df-so 5487 df-fr 5527 df-we 5529 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-pred 6179 df-ord 6237 df-on 6238 df-lim 6239 df-suc 6240 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-f1 6406 df-fo 6407 df-f1o 6408 df-fv 6409 df-riota 7192 df-ov 7238 df-oprab 7239 df-mpo 7240 df-om 7667 df-1st 7783 df-2nd 7784 df-wrecs 8071 df-recs 8132 df-rdg 8170 df-1o 8226 df-er 8415 df-en 8651 df-dom 8652 df-sdom 8653 df-fin 8654 df-card 9585 df-pnf 10899 df-mnf 10900 df-xr 10901 df-ltxr 10902 df-le 10903 df-sub 11094 df-neg 11095 df-nn 11861 df-2 11923 df-n0 12121 df-xnn0 12193 df-z 12207 df-uz 12469 df-fz 13126 df-hash 13930 df-upgr 27205 df-umgr 27206 |
This theorem is referenced by: vdumgr0 27600 vtxdumgrval 27606 umgrn1cycl 27923 upgracycumgr 32859 |
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