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Mirrors > Home > MPE Home > Th. List > relpths | Structured version Visualization version GIF version |
Description: The set (Paths‘𝐺) of all paths on 𝐺 is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
relpths | ⊢ Rel (Paths‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pths 28963 | . 2 ⊢ Paths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}) | |
2 | 1 | relmptopab 7653 | 1 ⊢ Rel (Paths‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 = wceq 1542 Vcvv 3475 ∩ cin 3947 ∅c0 4322 {cpr 4630 class class class wbr 5148 ◡ccnv 5675 ↾ cres 5678 “ cima 5679 Rel wrel 5681 Fun wfun 6535 ‘cfv 6541 (class class class)co 7406 0cc0 11107 1c1 11108 ..^cfzo 13624 ♯chash 14287 Trailsctrls 28937 Pathscpths 28959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fv 6549 df-pths 28963 |
This theorem is referenced by: iscycl 29038 cyclnspth 29047 |
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