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Theorem wrdval 13860

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.)

**Assertion**
Ref	Expression
wrdval	⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)))

Distinct variable groups: 𝑆,𝑙 𝑉,𝑙

Proof of Theorem wrdval Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-word 13858	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
2		eliun 4885	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_m (0..^𝑙)))
3		ovex 7168	. . . . . 6 ⊢ (0..^𝑙) ∈ V
4		elmapg 8402	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
5	3, 4	mpan2 690	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
6	5	rexbidv 3256	. . . 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 2927	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆})
9	1, 8	eqtr4id 2852	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 Vcvv 3441 ∪ ciun 4881 ⟶wf 6320 (class class class)co 7135 ↑_m cmap 8389 0cc0 10526 ℕ₀cn0 11885 ..^cfzo 13028 Word cword 13857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-word 13858

This theorem is referenced by: wrdexg 13867

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