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Mirrors > Home > MPE Home > Th. List > wspthnon | Structured version Visualization version GIF version |
Description: An element of the set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 15-Mar-2022.) |
Ref | Expression |
---|---|
wspthnon | ⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4927 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤 ↔ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)) | |
2 | 1 | exbidv 1880 | . 2 ⊢ (𝑤 = 𝑊 → (∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤 ↔ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)) |
3 | eqid 2772 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | 3 | iswspthsnon 27332 | . 2 ⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} |
5 | 2, 4 | elrab2 3593 | 1 ⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2048 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 Vtxcvtx 26474 SPathsOncspthson 27194 WWalksNOn cwwlksnon 27303 WSPathsNOn cwwspthsnon 27305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-wwlksnon 27308 df-wspthsnon 27310 |
This theorem is referenced by: wspthnonp 27335 wspthsnwspthsnon 27412 elwspths2on 27456 elwspths2spth 27463 |
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