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Mirrors > Home > MPE Home > Th. List > nbgraop1 | Structured version Visualization version GIF version |
Description: The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) |
Ref | Expression |
---|---|
nbgraop1 | ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (Fun 𝐸 → (〈𝑉, 𝐸〉 Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸‘𝑖) = {𝑁, 𝑛}})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgraop 25952 | . . 3 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (〈𝑉, 𝐸〉 Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}) | |
2 | elrnrexdmb 6272 | . . . . 5 ⊢ (Fun 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸 ↔ ∃𝑖 ∈ dom 𝐸{𝑁, 𝑛} = (𝐸‘𝑖))) | |
3 | eqcom 2617 | . . . . . 6 ⊢ ({𝑁, 𝑛} = (𝐸‘𝑖) ↔ (𝐸‘𝑖) = {𝑁, 𝑛}) | |
4 | 3 | rexbii 3023 | . . . . 5 ⊢ (∃𝑖 ∈ dom 𝐸{𝑁, 𝑛} = (𝐸‘𝑖) ↔ ∃𝑖 ∈ dom 𝐸(𝐸‘𝑖) = {𝑁, 𝑛}) |
5 | 2, 4 | syl6bb 275 | . . . 4 ⊢ (Fun 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸 ↔ ∃𝑖 ∈ dom 𝐸(𝐸‘𝑖) = {𝑁, 𝑛})) |
6 | 5 | rabbidv 3164 | . . 3 ⊢ (Fun 𝐸 → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸‘𝑖) = {𝑁, 𝑛}}) |
7 | 1, 6 | sylan9eq 2664 | . 2 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ Fun 𝐸) → (〈𝑉, 𝐸〉 Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸‘𝑖) = {𝑁, 𝑛}}) |
8 | 7 | ex 449 | 1 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (Fun 𝐸 → (〈𝑉, 𝐸〉 Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸‘𝑖) = {𝑁, 𝑛}})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 {cpr 4127 〈cop 4131 dom cdm 5038 ran crn 5039 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 Neighbors cnbgra 25946 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-nbgra 25949 |
This theorem is referenced by: (None) |
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