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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 | eliun 4524 | . . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑𝑚 (0..^𝑙))) | |
2 | ovex 6678 | . . . . . 6 ⊢ (0..^𝑙) ∈ V | |
3 | elmapg 7870 | . . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆)) | |
4 | 2, 3 | mpan2 707 | . . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆)) |
5 | 4 | rexbidv 3052 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆)) |
6 | 1, 5 | syl5bb 272 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆)) |
7 | 6 | abbi2dv 2742 | . 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}) |
8 | df-word 13299 | . 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
9 | 7, 8 | syl6reqr 2675 | 1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 Vcvv 3200 ∪ ciun 4520 ⟶wf 5884 (class class class)co 6650 ↑𝑚 cmap 7857 0cc0 9936 ℕ0cn0 11292 ..^cfzo 12465 Word cword 13291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-word 13299 |
This theorem is referenced by: wrdexg 13315 |
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