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Theorem nfixp1 8936
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 8916 . 2 X𝑥𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
2 nfcv 2892 . . . . 5 𝑥𝑦
3 nfab1 2894 . . . . 5 𝑥{𝑥𝑥𝐴}
42, 3nffn 6648 . . . 4 𝑥 𝑦 Fn {𝑥𝑥𝐴}
5 nfra1 3272 . . . 4 𝑥𝑥𝐴 (𝑦𝑥) ∈ 𝐵
64, 5nfan 1895 . . 3 𝑥(𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)
76nfab 2898 . 2 𝑥{𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
81, 7nfcxfr 2890 1 𝑥X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 394  wcel 2099  {cab 2703  wnfc 2876  wral 3051   Fn wfn 6538  cfv 6543  Xcixp 8915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5144  df-opab 5206  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-fun 6545  df-fn 6546  df-ixp 8916
This theorem is referenced by:  ixpiunwdom  9623  ptbasfi  23570  hoidmvlelem3  46251  hspdifhsp  46270  hoiqssbllem2  46277  hspmbllem2  46281  opnvonmbllem2  46287  iinhoiicc  46328  iunhoiioo  46330  vonioo  46336  vonicc  46339
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