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Theorem nfixp1 8090
Description: The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
Assertion
Ref Expression
nfixp1 𝑥X𝑥𝐴 𝐵

Proof of Theorem nfixp1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-ixp 8071 . 2 X𝑥𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
2 nfcv 2898 . . . . 5 𝑥𝑦
3 nfab1 2900 . . . . 5 𝑥{𝑥𝑥𝐴}
42, 3nffn 6144 . . . 4 𝑥 𝑦 Fn {𝑥𝑥𝐴}
5 nfra1 3075 . . . 4 𝑥𝑥𝐴 (𝑦𝑥) ∈ 𝐵
64, 5nfan 1973 . . 3 𝑥(𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)
76nfab 2903 . 2 𝑥{𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
81, 7nfcxfr 2896 1 𝑥X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 383  wcel 2135  {cab 2742  wnfc 2885  wral 3046   Fn wfn 6040  cfv 6045  Xcixp 8070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4801  df-opab 4861  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-fun 6047  df-fn 6048  df-ixp 8071
This theorem is referenced by:  ixpiunwdom  8657  ptbasfi  21582  hoidmvlelem3  41313  hspdifhsp  41332  hoiqssbllem2  41339  hspmbllem2  41343  opnvonmbllem2  41349  iinhoiicc  41390  iunhoiioo  41392  vonioo  41398  vonicc  41401
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