Theorem nfwrd 13333

Description: Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)

Hypothesis

Ref	Expression
nfwrd.1	⊢ Ⅎ𝑥𝑆

Assertion

Ref	Expression
nfwrd	⊢ Ⅎ𝑥Word 𝑆

Proof of Theorem nfwrd

Dummy variables 𝑤 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-word 13299	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥ℕ0
3		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑥𝑤
4		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑥(0..^𝑙)
5		nfwrd.1	. . . . 5 ⊢ Ⅎ𝑥𝑆
6	3, 4, 5	nff 6041	. . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆
7	2, 6	nfrex 3007	. . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
8	7	nfab 2769	. 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
9	1, 8	nfcxfr 2762	1 ⊢ Ⅎ𝑥Word 𝑆

Syntax hints:  {cab 2608  Ⅎwnfc 2751  ∃wrex 2913  ⟶wf 5884  (class class class)co 6650  0cc0 9936  ℕ₀cn0 11292  ..^cfzo 12465  Word cword 13291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-word 13299

This theorem is referenced by: (None)