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Mirrors > Home > MPE Home > Th. List > 2wot2wont | Structured version Visualization version GIF version |
Description: The set of (simple) paths of length 2 (in a graph) is the set of (simple) paths of length 2 between any two different vertices. (Contributed by Alexander van der Vekens, 27-Feb-2018.) |
Ref | Expression |
---|---|
2wot2wont | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 2WalksOt 𝐸) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ 𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el2wlksoton 26405 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑤 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))) | |
2 | vex 3176 | . . . . 5 ⊢ 𝑤 ∈ V | |
3 | eleq1 2676 | . . . . . 6 ⊢ (𝑢 = 𝑤 → (𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦) ↔ 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))) | |
4 | 3 | 2rexbidv 3039 | . . . . 5 ⊢ (𝑢 = 𝑤 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))) |
5 | 2, 4 | elab 3319 | . . . 4 ⊢ (𝑤 ∈ {𝑢 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)} ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)) |
6 | 1, 5 | syl6bbr 277 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑤 ∈ (𝑉 2WalksOt 𝐸) ↔ 𝑤 ∈ {𝑢 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)})) |
7 | 6 | eqrdv 2608 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 2WalksOt 𝐸) = {𝑢 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)}) |
8 | dfiunv2 4492 | . 2 ⊢ ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ 𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦) = {𝑢 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)} | |
9 | 7, 8 | syl6eqr 2662 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 2WalksOt 𝐸) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ 𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 ∪ ciun 4455 (class class class)co 6549 2WalksOt c2wlkot 26381 2WalksOnOt c2wlkonot 26382 |
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-rep 4699 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-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-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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 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-res 5050 df-ima 5051 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-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-2wlkonot 26385 df-2wlksot 26386 |
This theorem is referenced by: (None) |
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