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Theorem nfwrd 13882
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 13850 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2 nfcv 2974 . . . 4 𝑥0
3 nfcv 2974 . . . . 5 𝑥𝑤
4 nfcv 2974 . . . . 5 𝑥(0..^𝑙)
5 nfwrd.1 . . . . 5 𝑥𝑆
63, 4, 5nff 6503 . . . 4 𝑥 𝑤:(0..^𝑙)⟶𝑆
72, 6nfrex 3306 . . 3 𝑥𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
87nfab 2981 . 2 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
91, 8nfcxfr 2972 1 𝑥Word 𝑆
Colors of variables: wff setvar class
Syntax hints:  {cab 2796  wnfc 2958  wrex 3136  wf 6344  (class class class)co 7145  0cc0 10525  0cn0 11885  ..^cfzo 13021  Word cword 13849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352  df-word 13850
This theorem is referenced by: (None)
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