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Mirrors > Home > MPE Home > Th. List > frgrncvvdeqlem10 | Structured version Visualization version GIF version |
Description: Lemma 10 for frgrncvvdeq 28088. (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.) |
Ref | Expression |
---|---|
frgrncvvdeq.v1 | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrncvvdeq.e | ⊢ 𝐸 = (Edg‘𝐺) |
frgrncvvdeq.nx | ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) |
frgrncvvdeq.ny | ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) |
frgrncvvdeq.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
frgrncvvdeq.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
frgrncvvdeq.ne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
frgrncvvdeq.xy | ⊢ (𝜑 → 𝑌 ∉ 𝐷) |
frgrncvvdeq.f | ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) |
frgrncvvdeq.a | ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
Ref | Expression |
---|---|
frgrncvvdeqlem10 | ⊢ (𝜑 → 𝐴:𝐷–1-1-onto→𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrncvvdeq.v1 | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frgrncvvdeq.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | frgrncvvdeq.nx | . . 3 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) | |
4 | frgrncvvdeq.ny | . . 3 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) | |
5 | frgrncvvdeq.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
6 | frgrncvvdeq.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | frgrncvvdeq.ne | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
8 | frgrncvvdeq.xy | . . 3 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
9 | frgrncvvdeq.f | . . 3 ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) | |
10 | frgrncvvdeq.a | . . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | frgrncvvdeqlem8 28085 | . 2 ⊢ (𝜑 → 𝐴:𝐷–1-1→𝑁) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | frgrncvvdeqlem9 28086 | . 2 ⊢ (𝜑 → 𝐴:𝐷–onto→𝑁) |
13 | df-f1o 6362 | . 2 ⊢ (𝐴:𝐷–1-1-onto→𝑁 ↔ (𝐴:𝐷–1-1→𝑁 ∧ 𝐴:𝐷–onto→𝑁)) | |
14 | 11, 12, 13 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐴:𝐷–1-1-onto→𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∉ wnel 3123 {cpr 4569 ↦ cmpt 5146 –1-1→wf1 6352 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 Vtxcvtx 26781 Edgcedg 26832 NeighbVtx cnbgr 27114 FriendGraph cfrgr 28037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 df-edg 26833 df-upgr 26867 df-umgr 26868 df-usgr 26936 df-nbgr 27115 df-frgr 28038 |
This theorem is referenced by: frgrncvvdeq 28088 |
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