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Mirrors > Home > MPE Home > Th. List > clwwlkfv | Structured version Visualization version GIF version |
Description: Lemma 2 for clwwlkf1o 28134: the value of function 𝐹. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.) |
Ref | Expression |
---|---|
clwwlkf1o.d | ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} |
clwwlkf1o.f | ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁)) |
Ref | Expression |
---|---|
clwwlkfv | ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7220 | . 2 ⊢ (𝑡 = 𝑊 → (𝑡 prefix 𝑁) = (𝑊 prefix 𝑁)) | |
2 | clwwlkf1o.f | . 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁)) | |
3 | ovex 7246 | . 2 ⊢ (𝑊 prefix 𝑁) ∈ V | |
4 | 1, 2, 3 | fvmpt 6818 | 1 ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {crab 3065 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 0cc0 10729 lastSclsw 14117 prefix cpfx 14235 WWalksN cwwlksn 27910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 |
This theorem is referenced by: clwwlkf1 28132 clwwlkfo 28133 |
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