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Mirrors > Home > MPE Home > Th. List > wlkop | Structured version Visualization version GIF version |
Description: A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
Ref | Expression |
---|---|
wlkop | ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwlk 27559 | . 2 ⊢ Rel (Walks‘𝐺) | |
2 | 1st2nd 7756 | . 2 ⊢ ((Rel (Walks‘𝐺) ∧ 𝑊 ∈ (Walks‘𝐺)) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
3 | 1, 2 | mpan 690 | 1 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 〈cop 4519 Rel wrel 5524 ‘cfv 6333 1st c1st 7705 2nd c2nd 7706 Walkscwlks 27530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fv 6341 df-1st 7707 df-2nd 7708 df-wlks 27533 |
This theorem is referenced by: wlkcpr 27562 wlkeq 27567 clwlkcompbp 27715 clwlkclwwlkflem 27933 wlkl0 28296 |
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