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Theorem nfwrd 14492
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 14464 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2 nfcv 2903 . . . 4 𝑥0
3 nfcv 2903 . . . . 5 𝑥𝑤
4 nfcv 2903 . . . . 5 𝑥(0..^𝑙)
5 nfwrd.1 . . . . 5 𝑥𝑆
63, 4, 5nff 6713 . . . 4 𝑥 𝑤:(0..^𝑙)⟶𝑆
72, 6nfrexw 3310 . . 3 𝑥𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
87nfab 2909 . 2 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
91, 8nfcxfr 2901 1 𝑥Word 𝑆
Colors of variables: wff setvar class
Syntax hints:  {cab 2709  wnfc 2883  wrex 3070  wf 6539  (class class class)co 7408  0cc0 11109  0cn0 12471  ..^cfzo 13626  Word cword 14463
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547  df-word 14464
This theorem is referenced by: (None)
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