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Theorem nfwrd 10932
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 10905 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2 nfcv 2336 . . . 4 𝑥0
3 nfcv 2336 . . . . 5 𝑥𝑤
4 nfcv 2336 . . . . 5 𝑥(0..^𝑙)
5 nfwrd.1 . . . . 5 𝑥𝑆
63, 4, 5nff 5392 . . . 4 𝑥 𝑤:(0..^𝑙)⟶𝑆
72, 6nfrexw 2533 . . 3 𝑥𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆
87nfab 2341 . 2 𝑥{𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
91, 8nfcxfr 2333 1 𝑥Word 𝑆
Colors of variables: wff set class
Syntax hints:  {cab 2179  wnfc 2323  wrex 2473  wf 5242  (class class class)co 5910  0cc0 7862  0cn0 9230  ..^cfzo 10198  Word cword 10904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-fun 5248  df-fn 5249  df-f 5250  df-word 10905
This theorem is referenced by: (None)
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