Theorem iswrd 14450

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 14449	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2	1	eleq2i 2829	. 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ 𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3		ovex 7401	. . . . 5 ⊢ (0..^𝑙) ∈ V
4		fex 7182	. . . . 5 ⊢ ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V)
5	3, 4	mpan2 692	. . . 4 ⊢ (𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V)
6	5	rexlimivw 3135	. . 3 ⊢ (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V)
7		feq1 6648	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆))
8	7	rexbidv 3162	. . 3 ⊢ (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆))
9	6, 8	elab3 3643	. 2 ⊢ (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
10	2, 9	bitri 275	1 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3442  ⟶wf 6496  (class class class)co 7368  0cc0 11038  ℕ₀cn0 12413  ..^cfzo 13582  Word cword 14448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-word 14449

This theorem is referenced by:  iswrdi  14452  wrdf  14453  cshword  14726  motcgrg  28628  chnsubseqword  47230