Theorem wrdval 14551

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 14549	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		eliun 4999	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑m (0..^𝑙)))
3		ovex 7463	. . . . . 6 ⊢ (0..^𝑙) ∈ V
4		elmapg 8877	. . . . . 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 3176	. . . 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 2872	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
9	1, 8	eqtr4id 2793	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1536   ∈ wcel 2105  {cab 2711  ∃wrex 3067  Vcvv 3477  ∪ ciun 4995  ⟶wf 6558  (class class class)co 7430   ↑_m cmap 8864  0cc0 11152  ℕ₀cn0 12523  ..^cfzo 13690  Word cword 14548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-word 14549

This theorem is referenced by:  wrdexg  14558