Theorem wrdval 14463

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 14461	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		eliun 4991	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆 ↑m (0..^𝑙)))
3		ovex 7434	. . . . . 6 ⊢ (0..^𝑙) ∈ V
4		elmapg 8828	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
5	3, 4	mpan2 688	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑤 ∈ (𝑆 ↑m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
6	5	rexbidv 3170	. . . 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 2859	. 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
9	1, 8	eqtr4id 2783	1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {cab 2701  ∃wrex 3062  Vcvv 3466  ∪ ciun 4987  ⟶wf 6529  (class class class)co 7401   ↑_m cmap 8815  0cc0 11105  ℕ₀cn0 12468  ..^cfzo 13623  Word cword 14460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-word 14461

This theorem is referenced by:  wrdexg  14470