Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > frgrancvvdeqlem6 | Structured version Visualization version GIF version |
Description: Lemma 6 for frgrancvvdeq 26569. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017.) |
Ref | Expression |
---|---|
frgrancvvdeq.nx | ⊢ 𝐷 = (〈𝑉, 𝐸〉 Neighbors 𝑋) |
frgrancvvdeq.ny | ⊢ 𝑁 = (〈𝑉, 𝐸〉 Neighbors 𝑌) |
frgrancvvdeq.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
frgrancvvdeq.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
frgrancvvdeq.ne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
frgrancvvdeq.xy | ⊢ (𝜑 → 𝑌 ∉ 𝐷) |
frgrancvvdeq.f | ⊢ (𝜑 → 𝑉 FriendGrph 𝐸) |
frgrancvvdeq.a | ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) |
Ref | Expression |
---|---|
frgrancvvdeqlem6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷) | |
2 | riotaex 6515 | . . . 4 ⊢ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸) ∈ V | |
3 | frgrancvvdeq.a | . . . . 5 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) | |
4 | 3 | fvmpt2 6200 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸) ∈ V) → (𝐴‘𝑥) = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) |
5 | 1, 2, 4 | sylancl 693 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) |
6 | 5 | sneqd 4137 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)}) |
7 | frgrancvvdeq.nx | . . 3 ⊢ 𝐷 = (〈𝑉, 𝐸〉 Neighbors 𝑋) | |
8 | frgrancvvdeq.ny | . . 3 ⊢ 𝑁 = (〈𝑉, 𝐸〉 Neighbors 𝑌) | |
9 | frgrancvvdeq.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
10 | frgrancvvdeq.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | frgrancvvdeq.ne | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
12 | frgrancvvdeq.xy | . . 3 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
13 | frgrancvvdeq.f | . . 3 ⊢ (𝜑 → 𝑉 FriendGrph 𝐸) | |
14 | 7, 8, 9, 10, 11, 12, 13, 3 | frgrancvvdeqlem4 26560 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁)) |
15 | 6, 14 | eqtrd 2644 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((〈𝑉, 𝐸〉 Neighbors 𝑥) ∩ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 Vcvv 3173 ∩ cin 3539 {csn 4125 {cpr 4127 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Neighbors cnbgra 25946 FriendGrph cfrgra 26515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-usgra 25862 df-nbgra 25949 df-frgra 26516 |
This theorem is referenced by: frgrancvvdeqlem7 26563 frgrancvvdeqlemA 26564 frgrancvvdeqlemC 26566 |
Copyright terms: Public domain | W3C validator |