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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgr3stgrlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for isubgr3stgr 47942. (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 47930 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘𝑊) = (𝑁 + 1)) |
| 5 | 1, 4 | ax-mp 5 | . 2 ⊢ (♯‘𝑊) = (𝑁 + 1) |
| 6 | oveq1 7438 | . . . . . 6 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((♯‘𝑊) − 1) = ((𝑁 + 1) − 1)) | |
| 7 | nn0cn 12536 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
| 8 | pncan1 11687 | . . . . . . . 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 2777 | . . . . 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 12566 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
| 14 | 1, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑁 + 1) ∈ ℕ0 |
| 15 | eleq1 2829 | . . . . . . . 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 14397 | . . . . . . 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 47929 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ 𝑊) |
| 23 | 1, 22 | ax-mp 5 | . . . . 5 ⊢ 0 ∈ 𝑊 |
| 24 | hashdifsn 14453 | . . . . 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 1197 | . . . 4 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (♯‘𝑈) = 𝑁) | |
| 27 | 12, 25, 26 | 3eqtr4rd 2788 | . . 3 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁)) → (♯‘𝑈) = (♯‘(𝑊 ∖ {0}))) |
| 28 | eleq1 2829 | . . . . . . 7 ⊢ ((♯‘𝑈) = 𝑁 → ((♯‘𝑈) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0)) | |
| 29 | 1, 28 | mpbiri 258 | . . . . . 6 ⊢ ((♯‘𝑈) = 𝑁 → (♯‘𝑈) ∈ ℕ0) |
| 30 | 29 | 3ad2ant3 1136 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ (♯‘𝑈) = 𝑁) → (♯‘𝑈) ∈ ℕ0) |
| 31 | isubgr3stgr.u | . . . . . . 7 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) | |
| 32 | 31 | ovexi 7465 | . . . . . 6 ⊢ 𝑈 ∈ V |
| 33 | hashclb 14397 | . . . . . 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 9215 | . . . . 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 14388 | . . . 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 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 {csn 4626 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 − cmin 11492 ℕ0cn0 12526 ♯chash 14369 Vtxcvtx 29013 USGraphcusgr 29166 NeighbVtx cnbgr 29349 ClNeighbVtx cclnbgr 47805 StarGrcstgr 47918 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-hash 14370 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-edgf 29004 df-vtx 29015 df-stgr 47919 |
| This theorem is referenced by: isubgr3stgrlem3 47935 |
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