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Theorem nfwrd 13280
Description: Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)
Hypothesis
Ref Expression
nfwrd.1 𝑥𝑆
Assertion
Ref Expression
nfwrd 𝑥Word 𝑆

Proof of Theorem nfwrd
Dummy variables 𝑤 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-word 13246 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2 nfcv 2761 . . . 4 𝑥0
3 nfcv 2761 . . . . 5 𝑥𝑤
4 nfcv 2761 . . . . 5 𝑥(0..^𝑙)
5 nfwrd.1 . . . . 5 𝑥𝑆
63, 4, 5nff 6003 . . . 4 𝑥 𝑤:(0..^𝑙)⟶𝑆
72, 6nfrex 3002 . . 3 𝑥𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
87nfab 2765 . 2 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
91, 8nfcxfr 2759 1 𝑥Word 𝑆
Colors of variables: wff setvar class
Syntax hints:  {cab 2607  wnfc 2748  wrex 2908  wf 5848  (class class class)co 6610  0cc0 9888  0cn0 11244  ..^cfzo 12414  Word cword 13238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-fun 5854  df-fn 5855  df-f 5856  df-word 13246
This theorem is referenced by: (None)
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