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Mirrors > Home > MPE Home > Th. List > cusgrfilem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for cusgrfi 29246. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
Ref | Expression |
---|---|
cusgrfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
cusgrfi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} |
Ref | Expression |
---|---|
cusgrfilem1 | ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cusgrfi.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2727 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | cusgredg 29211 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
4 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑥 = {𝑎, 𝑁} → (♯‘𝑥) = (♯‘{𝑎, 𝑁})) | |
5 | 4 | ad2antlr 726 | . . . . . . . 8 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (♯‘𝑥) = (♯‘{𝑎, 𝑁})) |
6 | hashprg 14372 | . . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑎 ≠ 𝑁 ↔ (♯‘{𝑎, 𝑁}) = 2)) | |
7 | 6 | adantrr 716 | . . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉)) → (𝑎 ≠ 𝑁 ↔ (♯‘{𝑎, 𝑁}) = 2)) |
8 | 7 | biimpcd 248 | . . . . . . . . . 10 ⊢ (𝑎 ≠ 𝑁 → ((𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉)) → (♯‘{𝑎, 𝑁}) = 2)) |
9 | 8 | adantr 480 | . . . . . . . . 9 ⊢ ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) → ((𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉)) → (♯‘{𝑎, 𝑁}) = 2)) |
10 | 9 | imp 406 | . . . . . . . 8 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (♯‘{𝑎, 𝑁}) = 2) |
11 | 5, 10 | eqtrd 2767 | . . . . . . 7 ⊢ (((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ∧ (𝑎 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉))) → (♯‘𝑥) = 2) |
12 | 11 | an13s 650 | . . . . . 6 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}))) → (♯‘𝑥) = 2) |
13 | 12 | rexlimdvaa 3151 | . . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑉) → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) → (♯‘𝑥) = 2)) |
14 | 13 | ss2rabdv 4069 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
15 | cusgrfi.p | . . . . . 6 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} | |
16 | 15 | a1i 11 | . . . . 5 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}) |
17 | id 22 | . . . . 5 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → (Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | |
18 | 16, 17 | sseq12d 4011 | . . . 4 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → (𝑃 ⊆ (Edg‘𝐺) ↔ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) |
19 | 14, 18 | imbitrrid 245 | . . 3 ⊢ ((Edg‘𝐺) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → (𝑁 ∈ 𝑉 → 𝑃 ⊆ (Edg‘𝐺))) |
20 | 3, 19 | syl 17 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → (𝑁 ∈ 𝑉 → 𝑃 ⊆ (Edg‘𝐺))) |
21 | 20 | imp 406 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑃 ⊆ (Edg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∃wrex 3065 {crab 3427 ⊆ wss 3944 𝒫 cpw 4598 {cpr 4626 ‘cfv 6542 2c2 12283 ♯chash 14307 Vtxcvtx 28783 Edgcedg 28834 ComplUSGraphccusgr 29197 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-xnn0 12561 df-z 12575 df-uz 12839 df-fz 13503 df-hash 14308 df-edg 28835 df-upgr 28869 df-umgr 28870 df-usgr 28938 df-nbgr 29120 df-uvtx 29173 df-cplgr 29198 df-cusgr 29199 |
This theorem is referenced by: cusgrfi 29246 |
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