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| Description: Lemma 4 for frgrncvvdeq 30328. The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 10-May-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 | 
|---|---|
| frgrncvvdeqlem4 | ⊢ (𝜑 → 𝐴:𝐷⟶𝑁) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | frgrncvvdeq.v1 | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | frgrncvvdeq.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
| 3 | frgrncvvdeq.nx | . . . 4 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) | |
| 4 | frgrncvvdeq.ny | . . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) | |
| 5 | frgrncvvdeq.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 6 | frgrncvvdeq.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 7 | frgrncvvdeq.ne | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
| 8 | frgrncvvdeq.xy | . . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
| 9 | frgrncvvdeq.f | . . . 4 ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) | |
| 10 | frgrncvvdeq.a | . . . 4 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) | |
| 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | frgrncvvdeqlem2 30319 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) | 
| 12 | riotacl 7405 | . . 3 ⊢ (∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) ∈ 𝑁) | |
| 13 | 11, 12 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) ∈ 𝑁) | 
| 14 | 13, 10 | fmptd 7134 | 1 ⊢ (𝜑 → 𝐴:𝐷⟶𝑁) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∃!wreu 3378 {cpr 4628 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Vtxcvtx 29013 Edgcedg 29064 NeighbVtx cnbgr 29349 FriendGraph cfrgr 30277 | 
| 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-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-rmo 3380 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-2o 8507 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-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-edg 29065 df-upgr 29099 df-umgr 29100 df-usgr 29168 df-nbgr 29350 df-frgr 30278 | 
| This theorem is referenced by: frgrncvvdeqlem8 30325 frgrncvvdeqlem9 30326 | 
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