Theorem nfwrd 14505

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 14476	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		nfcv 2898	. . . 4 ⊢ Ⅎ𝑥ℕ0
3		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑤
4		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥(0..^𝑙)
5		nfwrd.1	. . . . 5 ⊢ Ⅎ𝑥𝑆
6	3, 4, 5	nff 6664	. . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆
7	2, 6	nfrexw 3285	. . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
8	7	nfab 2904	. 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
9	1, 8	nfcxfr 2896	1 ⊢ Ⅎ𝑥Word 𝑆

Syntax hints:  {cab 2714  Ⅎwnfc 2883  ∃wrex 3061  ⟶wf 6494  (class class class)co 7367  0cc0 11038  ℕ₀cn0 12437  ..^cfzo 13608  Word cword 14475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6500  df-fn 6501  df-f 6502  df-word 14476

This theorem is referenced by: (None)