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Theorem wrdval 14148

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 14146	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
2		eliun 4925	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_m (0..^𝑙)))
3		ovex 7288	. . . . . 6 ⊢ (0..^𝑙) ∈ V
4		elmapg 8586	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
5	3, 4	mpan2 687	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
6	5	rexbidv 3225	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ₀ 𝑤 ∈ (𝑆 ↑_m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆))
7	2, 6	syl5bb 282	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆))
8	7	abbi2dv 2876	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆})
9	1, 8	eqtr4id 2798	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2108 {cab 2715 ∃wrex 3064 Vcvv 3422 ∪ ciun 4921 ⟶wf 6414 (class class class)co 7255 ↑_m cmap 8573 0cc0 10802 ℕ₀cn0 12163 ..^cfzo 13311 Word cword 14145

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-word 14146

This theorem is referenced by: wrdexg 14155

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