Theorem nfwrd 14578

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 14550	. 2 ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2		nfcv 2903	. . . 4 ⊢ Ⅎ𝑥ℕ0
3		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝑤
4		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥(0..^𝑙)
5		nfwrd.1	. . . . 5 ⊢ Ⅎ𝑥𝑆
6	3, 4, 5	nff 6733	. . . 4 ⊢ Ⅎ𝑥 𝑤:(0..^𝑙)⟶𝑆
7	2, 6	nfrexw 3311	. . 3 ⊢ Ⅎ𝑥∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
8	7	nfab 2909	. 2 ⊢ Ⅎ𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
9	1, 8	nfcxfr 2901	1 ⊢ Ⅎ𝑥Word 𝑆

Syntax hints:  {cab 2712  Ⅎwnfc 2888  ∃wrex 3068  ⟶wf 6559  (class class class)co 7431  0cc0 11153  ℕ₀cn0 12524  ..^cfzo 13691  Word cword 14549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-word 14550

This theorem is referenced by: (None)