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Theorem wrdval 13867

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		eliun 4925	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_m (0..^𝑙)))
2		ovex 7191	. . . . . 6 ⊢ (0..^𝑙) ∈ V
3		elmapg 8421	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
4	2, 3	mpan2 689	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
5	4	rexbidv 3299	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆))
6	1, 5	syl5bb 285	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆))
7	6	abbi2dv 2952	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆})
8		df-word 13865	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
9	7, 8	syl6reqr 2877	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 Vcvv 3496 ∪ ciun 4921 ⟶wf 6353 (class class class)co 7158 ↑_m cmap 8408 0cc0 10539 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 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-word 13865

This theorem is referenced by: wrdexg 13874 wrdexgOLD 13875

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