Theorem iswrd 14542

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 14541	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2	1	eleq2i 2857	. 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ 𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3		ovex 7433	. . . . 5 ⊢ (0..^𝑙) ∈ V
4		fex 7214	. . . . 5 ⊢ ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V)
5	3, 4	mpan2 703	. . . 4 ⊢ (𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V)
6	5	rexlimivw 3162	. . 3 ⊢ (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆 → 𝑊 ∈ V)
7		feq1 6673	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆 ↔ 𝑊:(0..^𝑙)⟶𝑆))
8	7	rexbidv 3189	. . 3 ⊢ (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆))
9	6, 8	elab3 3648	. 2 ⊢ (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
10	2, 9	bitri 278	1 ⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)

Syntax hints:   ↔ wb 209   = wceq 1563   ∈ wcel 2145  {cab 2743  ∃wrex 3089  Vcvv 3457  ⟶wf 6521  (class class class)co 7400  0cc0 11088  ℕ₀cn0 12495  ..^cfzo 13673  Word cword 14540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-word 14541

This theorem is referenced by:  iswrdi  14544  wrdf  14545  cshword  14818  motcgrg  28771  chnsubseqword  47452