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Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgr3stgrlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for isubgr3stgr 47877. (Contributed by AV, 16-Sep-2025.) |
Ref | Expression |
---|---|
isubgr3stgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isubgr3stgr.u | ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) |
isubgr3stgr.c | ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋) |
isubgr3stgr.n | ⊢ 𝑁 ∈ ℕ0 |
isubgr3stgr.s | ⊢ 𝑆 = (StarGr‘𝑁) |
isubgr3stgr.w | ⊢ 𝑊 = (Vtx‘𝑆) |
Ref | Expression |
---|---|
isubgr3stgrlem2 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isubgr3stgr.n | . . 3 ⊢ 𝑁 ∈ ℕ0 | |
2 | isubgr3stgr.s | . . . 4 ⊢ 𝑆 = (StarGr‘𝑁) | |
3 | isubgr3stgr.w | . . . 4 ⊢ 𝑊 = (Vtx‘𝑆) | |
4 | 2, 3 | stgrorder 47865 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘𝑊) = (𝑁 + 1)) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ (♯‘𝑊) = (𝑁 + 1) |
6 | oveq1 7437 | . . . . . 6 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((♯‘𝑊) − 1) = ((𝑁 + 1) − 1)) | |
7 | nn0cn 12533 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
8 | pncan1 11684 | . . . . . . . 8 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁) |
10 | 1, 9 | mp1i 13 | . . . . . 6 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((𝑁 + 1) − 1) = 𝑁) |
11 | 6, 10 | eqtrd 2774 | . . . . 5 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((♯‘𝑊) − 1) = 𝑁) |
12 | 11 | adantr 480 | . . . 4 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → ((♯‘𝑊) − 1) = 𝑁) |
13 | peano2nn0 12563 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
14 | 1, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑁 + 1) ∈ ℕ0 |
15 | eleq1 2826 | . . . . . . . 8 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((♯‘𝑊) ∈ ℕ0 ↔ (𝑁 + 1) ∈ ℕ0)) | |
16 | 14, 15 | mpbiri 258 | . . . . . . 7 ⊢ ((♯‘𝑊) = (𝑁 + 1) → (♯‘𝑊) ∈ ℕ0) |
17 | 16 | adantr 480 | . . . . . 6 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (♯‘𝑊) ∈ ℕ0) |
18 | 3 | fvexi 6920 | . . . . . . 7 ⊢ 𝑊 ∈ V |
19 | hashclb 14393 | . . . . . . 7 ⊢ (𝑊 ∈ V → (𝑊 ∈ Fin ↔ (♯‘𝑊) ∈ ℕ0)) | |
20 | 18, 19 | mp1i 13 | . . . . . 6 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (𝑊 ∈ Fin ↔ (♯‘𝑊) ∈ ℕ0)) |
21 | 17, 20 | mpbird 257 | . . . . 5 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → 𝑊 ∈ Fin) |
22 | 2, 3 | stgrvtx0 47864 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑊) |
23 | 1, 22 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ 𝑊 |
24 | hashdifsn 14449 | . . . . 5 ⊢ ((𝑊 ∈ Fin ∧ 0 ∈ 𝑊) → (♯‘(𝑊 ∖ {0})) = ((♯‘𝑊) − 1)) | |
25 | 21, 23, 24 | sylancl 586 | . . . 4 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (♯‘(𝑊 ∖ {0})) = ((♯‘𝑊) − 1)) |
26 | simpr3 1195 | . . . 4 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (♯‘𝑈) = 𝑁) | |
27 | 12, 25, 26 | 3eqtr4rd 2785 | . . 3 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (♯‘𝑈) = (♯‘(𝑊 ∖ {0}))) |
28 | eleq1 2826 | . . . . . . 7 ⊢ ((♯‘𝑈) = 𝑁 → ((♯‘𝑈) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0)) | |
29 | 1, 28 | mpbiri 258 | . . . . . 6 ⊢ ((♯‘𝑈) = 𝑁 → (♯‘𝑈) ∈ ℕ0) |
30 | 29 | 3ad2ant3 1134 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (♯‘𝑈) ∈ ℕ0) |
31 | isubgr3stgr.u | . . . . . . 7 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) | |
32 | 31 | ovexi 7464 | . . . . . 6 ⊢ 𝑈 ∈ V |
33 | hashclb 14393 | . . . . . 6 ⊢ (𝑈 ∈ V → (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ0)) | |
34 | 32, 33 | mp1i 13 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (𝑈 ∈ Fin ↔ (♯‘𝑈) ∈ ℕ0)) |
35 | 30, 34 | mpbird 257 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → 𝑈 ∈ Fin) |
36 | diffi 9213 | . . . . 5 ⊢ (𝑊 ∈ Fin → (𝑊 ∖ {0}) ∈ Fin) | |
37 | 21, 36 | syl 17 | . . . 4 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (𝑊 ∖ {0}) ∈ Fin) |
38 | hasheqf1o 14384 | . . . 4 ⊢ ((𝑈 ∈ Fin ∧ (𝑊 ∖ {0}) ∈ Fin) → ((♯‘𝑈) = (♯‘(𝑊 ∖ {0})) ↔ ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))) | |
39 | 35, 37, 38 | syl2an2 686 | . . 3 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → ((♯‘𝑈) = (♯‘(𝑊 ∖ {0})) ↔ ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0}))) |
40 | 27, 39 | mpbid 232 | . 2 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) |
41 | 5, 40 | mpan 690 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈–1-1-onto→(𝑊 ∖ {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 {csn 4630 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 − cmin 11489 ℕ0cn0 12523 ♯chash 14365 Vtxcvtx 29027 USGraphcusgr 29180 NeighbVtx cnbgr 29363 ClNeighbVtx cclnbgr 47742 StarGrcstgr 47853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-hash 14366 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-edgf 29018 df-vtx 29029 df-stgr 47854 |
This theorem is referenced by: isubgr3stgrlem3 47870 |
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