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Mirrors > Home > MPE Home > Th. List > wrdval | Structured version Visualization version GIF version |
Description: Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
wrdval | ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-word 13968 | . 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
2 | eliun 4895 | . . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑m (0..^𝑙))) | |
3 | ovex 7215 | . . . . . 6 ⊢ (0..^𝑙) ∈ V | |
4 | elmapg 8462 | . . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆)) | |
5 | 3, 4 | mpan2 691 | . . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆)) |
6 | 5 | rexbidv 3208 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆)) |
7 | 2, 6 | syl5bb 286 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆)) |
8 | 7 | abbi2dv 2870 | . 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}) |
9 | 1, 8 | eqtr4id 2793 | 1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2114 {cab 2717 ∃wrex 3055 Vcvv 3400 ∪ ciun 4891 ⟶wf 6345 (class class class)co 7182 ↑m cmap 8449 0cc0 10627 ℕ0cn0 11988 ..^cfzo 13136 Word cword 13967 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
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 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7185 df-oprab 7186 df-mpo 7187 df-map 8451 df-word 13968 |
This theorem is referenced by: wrdexg 13977 |
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