Theorem iswrd 14554

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 14553	. . 3 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2	1	eleq2i 2855	. 2 ⊢ (𝑊 ∈ Word 𝑆 ↔ 𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3		ovex 7445	. . . . 5 ⊢ (0..^𝑙) ∈ V
4		fex 7226	. . . . 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 6685	. . . 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 3646	. 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 1570   ∈ wcel 2143  {cab 2741  ∃wrex 3089  Vcvv 3455  ⟶wf 6534  (class class class)co 7412  0cc0 11101  ℕ₀cn0 12505  ..^cfzo 13684  Word cword 14552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-word 14553

This theorem is referenced by:  iswrdi  14556  wrdf  14557  cshword  14830  motcgrg  28791  chnsubseqword  47574