Theorem nfwrd 14580

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 14551	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		nfcv 2931	. . . 4 ⊢ Ⅎ𝑥ℕ0
3		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑥𝑤
4		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑥(0..^𝑙)
5		nfwrd.1	. . . . 5 ⊢ Ⅎ𝑥𝑆
6	3, 4, 5	nff 6702	. . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆
7	2, 6	nfrexw 3319	. . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
8	7	nfab 2937	. 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
9	1, 8	nfcxfr 2929	1 ⊢ Ⅎ𝑥Word 𝑆

Syntax hints:  {cab 2747  Ⅎwnfc 2916  ∃wrex 3095  ⟶wf 6533  (class class class)co 7411  0cc0 11100  ℕ₀cn0 12504  ..^cfzo 13682  Word cword 14550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-fun 6539  df-fn 6540  df-f 6541  df-word 14551

This theorem is referenced by: (None)