Theorem wrdval 14425

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 14423	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		eliun 4945	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑m (0..^𝑙)))
3		ovex 7385	. . . . . 6 ⊢ (0..^𝑙) ∈ V
4		elmapg 8769	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
5	3, 4	mpan2 691	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
6	5	rexbidv 3157	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
7	2, 6	bitrid 283	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
8	7	eqabdv 2866	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
9	1, 8	eqtr4id 2787	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {cab 2711  ∃wrex 3057  Vcvv 3437  ∪ ciun 4941  ⟶wf 6482  (class class class)co 7352   ↑_m cmap 8756  0cc0 11013  ℕ₀cn0 12388  ..^cfzo 13556  Word cword 14422

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758  df-word 14423

This theorem is referenced by:  wrdexg  14433