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Theorem wrdval 13537

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 4714	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_𝑚 (0..^𝑙)))
2		ovex 6910	. . . . . 6 ⊢ (0..^𝑙) ∈ V
3		elmapg 8108	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
4	2, 3	mpan2 683	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
5	4	rexbidv 3233	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆))
6	1, 5	syl5bb 275	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆))
7	6	abbi2dv 2919	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆})
8		df-word 13535	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
9	7, 8	syl6reqr 2852	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 {cab 2785 ∃wrex 3090 Vcvv 3385 ∪ ciun 4710 ⟶wf 6097 (class class class)co 6878 ↑_𝑚 cmap 8095 0cc0 10224 ℕ₀cn0 11580 ..^cfzo 12720 Word cword 13534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-map 8097 df-word 13535

This theorem is referenced by: wrdexg 13544

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