Theorem nfwrd 13519

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 13485	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		nfcv 2902	. . . 4 ⊢ Ⅎ𝑥ℕ0
3		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑤
4		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥(0..^𝑙)
5		nfwrd.1	. . . . 5 ⊢ Ⅎ𝑥𝑆
6	3, 4, 5	nff 6202	. . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆
7	2, 6	nfrex 3145	. . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
8	7	nfab 2907	. 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
9	1, 8	nfcxfr 2900	1 ⊢ Ⅎ𝑥Word 𝑆

Syntax hints:  {cab 2746  Ⅎwnfc 2889  ∃wrex 3051  ⟶wf 6045  (class class class)co 6813  0cc0 10128  ℕ₀cn0 11484  ..^cfzo 12659  Word cword 13477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-fun 6051  df-fn 6052  df-f 6053  df-word 13485

This theorem is referenced by: (None)