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Theorem wrdval 13494

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 4676	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_𝑚 (0..^𝑙)))
2		ovex 6841	. . . . . 6 ⊢ (0..^𝑙) ∈ V
3		elmapg 8036	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
4	2, 3	mpan2 709	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑_𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
5	4	rexbidv 3190	. . . 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 2880	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆})
8		df-word 13485	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
9	7, 8	syl6reqr 2813	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 = wceq 1632 ∈ wcel 2139 {cab 2746 ∃wrex 3051 Vcvv 3340 ∪ ciun 4672 ⟶wf 6045 (class class class)co 6813 ↑_𝑚 cmap 8023 0cc0 10128 ℕ₀cn0 11484 ..^cfzo 12659 Word cword 13477

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-map 8025 df-word 13485

This theorem is referenced by: wrdexg 13501

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