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Mirrors > Home > MPE Home > Th. List > wspniunwspnon | Structured version Visualization version GIF version |
Description: The set of nonempty simple paths of fixed length is the double union of the simple paths of the fixed length between different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 15-Mar-2022.) |
Ref | Expression |
---|---|
wspniunwspnon.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wspniunwspnon | ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wspthsnonn0vne 29947 | . . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ≠ ∅) → 𝑥 ≠ 𝑦) | |
2 | 1 | ex 412 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ≠ ∅ → 𝑥 ≠ 𝑦)) |
3 | 2 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → ((𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ≠ ∅ → 𝑥 ≠ 𝑦)) |
4 | ne0i 4347 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) → (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ≠ ∅) | |
5 | 3, 4 | impel 505 | . . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) ∧ 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)) → 𝑥 ≠ 𝑦) |
6 | 5 | necomd 2994 | . . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) ∧ 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)) → 𝑦 ≠ 𝑥) |
7 | 6 | ex 412 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) → 𝑦 ≠ 𝑥)) |
8 | 7 | pm4.71rd 562 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ (𝑦 ≠ 𝑥 ∧ 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)))) |
9 | 8 | rexbidv 3177 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (∃𝑦 ∈ 𝑉 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)))) |
10 | rexdifsn 4799 | . . . . . 6 ⊢ (∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑦 ∈ 𝑉 (𝑦 ≠ 𝑥 ∧ 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦))) | |
11 | 9, 10 | bitr4di 289 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (∃𝑦 ∈ 𝑉 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦))) |
12 | 11 | rexbidv 3177 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦))) |
13 | wspniunwspnon.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
14 | 13 | wspthsnwspthsnon 29946 | . . . 4 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)) |
15 | vex 3482 | . . . . 5 ⊢ 𝑤 ∈ V | |
16 | eleq1w 2822 | . . . . . . 7 ⊢ (𝑝 = 𝑤 → (𝑝 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ 𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦))) | |
17 | 16 | rexbidv 3177 | . . . . . 6 ⊢ (𝑝 = 𝑤 → (∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑝 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦))) |
18 | 17 | rexbidv 3177 | . . . . 5 ⊢ (𝑝 = 𝑤 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑝 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦))) |
19 | 15, 18 | elab 3681 | . . . 4 ⊢ (𝑤 ∈ {𝑝 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑝 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)} ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑤 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)) |
20 | 12, 14, 19 | 3bitr4g 314 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑤 ∈ (𝑁 WSPathsN 𝐺) ↔ 𝑤 ∈ {𝑝 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑝 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)})) |
21 | 20 | eqrdv 2733 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsN 𝐺) = {𝑝 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑝 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)}) |
22 | dfiunv2 5040 | . 2 ⊢ ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦) = {𝑝 ∣ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})𝑝 ∈ (𝑥(𝑁 WSPathsNOn 𝐺)𝑦)} | |
23 | 21, 22 | eqtr4di 2793 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ≠ wne 2938 ∃wrex 3068 ∖ cdif 3960 ∅c0 4339 {csn 4631 ∪ ciun 4996 ‘cfv 6563 (class class class)co 7431 ℕcn 12264 Vtxcvtx 29028 WSPathsN cwwspthsn 29858 WSPathsNOn cwwspthsnon 29859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-wlks 29632 df-wlkson 29633 df-trls 29725 df-trlson 29726 df-pths 29749 df-spths 29750 df-spthson 29752 df-wwlks 29860 df-wwlksn 29861 df-wwlksnon 29862 df-wspthsn 29863 df-wspthsnon 29864 |
This theorem is referenced by: frgrhash2wsp 30361 |
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