Theorem iswrd 14147

Description: Property of being a word over a set with an existential quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

Assertion

Ref	Expression
iswrd	⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)

Distinct variable groups:   𝑆,𝑙   𝑊,𝑙

Proof of Theorem iswrd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-word 14146	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2	1	eleq2i 2830	. 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ 𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3		ovex 7288	. . . . 5 ⊢ (0..^𝑙) ∈ V
4		fex 7084	. . . . 5 ⊢ ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V)
5	3, 4	mpan2 687	. . . 4 ⊢ (𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V)
6	5	rexlimivw 3210	. . 3 ⊢ (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V)
7		feq1 6565	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆))
8	7	rexbidv 3225	. . 3 ⊢ (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆))
9	6, 8	elab3 3610	. 2 ⊢ (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
10	2, 9	bitri 274	1 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422  ⟶wf 6414  (class class class)co 7255  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-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

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-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-word 14146

This theorem is referenced by:  iswrdi  14149  wrdf  14150  cshword  14432  motcgrg  26809