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Mirrors > Home > MPE Home > Th. List > 2clwwlkel | Structured version Visualization version GIF version |
Description: Characterization of an element of the value of operation 𝐶, i.e., of a word being a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.) |
Ref | Expression |
---|---|
2clwwlk.c | ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
Ref | Expression |
---|---|
2clwwlkel | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2clwwlk.c | . . . 4 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) | |
2 | 1 | 2clwwlk 30229 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
3 | 2 | eleq2d 2811 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})) |
4 | fveq1 6895 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤‘(𝑁 − 2)) = (𝑊‘(𝑁 − 2))) | |
5 | 4 | eqeq1d 2727 | . . 3 ⊢ (𝑤 = 𝑊 → ((𝑤‘(𝑁 − 2)) = 𝑋 ↔ (𝑊‘(𝑁 − 2)) = 𝑋)) |
6 | 5 | elrab 3679 | . 2 ⊢ (𝑊 ∈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋)) |
7 | 3, 6 | bitrdi 286 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3418 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 − cmin 11476 2c2 12300 ℤ≥cuz 12855 ClWWalksNOncclwwlknon 29969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 |
This theorem is referenced by: 2clwwlk2clwwlk 30232 numclwwlk1lem2f1 30239 |
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