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Mirrors > Home > MPE Home > Th. List > iswrd | Structured version Visualization version GIF version |
Description: Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.) |
Ref | Expression |
---|---|
iswrd | ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-word 13865 | . . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} | |
2 | 1 | eleq2i 2907 | . 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ 𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}) |
3 | ovex 7192 | . . . . 5 ⊢ (0..^𝑙) ∈ V | |
4 | fex 6992 | . . . . 5 ⊢ ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V) | |
5 | 3, 4 | mpan2 689 | . . . 4 ⊢ (𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V) |
6 | 5 | rexlimivw 3285 | . . 3 ⊢ (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V) |
7 | feq1 6498 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆)) | |
8 | 7 | rexbidv 3300 | . . 3 ⊢ (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)) |
9 | 6, 8 | elab3 3677 | . 2 ⊢ (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
10 | 2, 9 | bitri 277 | 1 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ∈ wcel 2113 {cab 2802 ∃wrex 3142 Vcvv 3497 ⟶wf 6354 (class class class)co 7159 0cc0 10540 ℕ0cn0 11900 ..^cfzo 13036 Word cword 13864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-word 13865 |
This theorem is referenced by: iswrdi 13868 wrdf 13869 cshword 14156 motcgrg 26333 |
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