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Mirrors > Home > MPE Home > Th. List > eupthi | Structured version Visualization version GIF version |
Description: Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
eupths.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
eupthi | ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupths.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | 1 | iseupthf1o 29149 | . 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)) |
3 | 2 | biimpi 215 | 1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 class class class wbr 5106 dom cdm 5634 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 0cc0 11052 ..^cfzo 13568 ♯chash 14231 iEdgciedg 27951 Walkscwlks 28547 EulerPathsceupth 29144 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-wlks 28550 df-trls 28643 df-eupth 29145 |
This theorem is referenced by: eupthf1o 29151 eupthseg 29153 eupthcl 29157 eupthp1 29163 |
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