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Theorem nfixp1 8790
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 8770 . 2 X𝑥𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
2 nfcv 2906 . . . . 5 𝑥𝑦
3 nfab1 2908 . . . . 5 𝑥{𝑥𝑥𝐴}
42, 3nffn 6597 . . . 4 𝑥 𝑦 Fn {𝑥𝑥𝐴}
5 nfra1 3266 . . . 4 𝑥𝑥𝐴 (𝑦𝑥) ∈ 𝐵
64, 5nfan 1903 . . 3 𝑥(𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)
76nfab 2912 . 2 𝑥{𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
81, 7nfcxfr 2904 1 𝑥X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 397  wcel 2107  {cab 2715  wnfc 2886  wral 3063   Fn wfn 6487  cfv 6492  Xcixp 8769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6494  df-fn 6495  df-ixp 8770
This theorem is referenced by:  ixpiunwdom  9460  ptbasfi  22854  hoidmvlelem3  44560  hspdifhsp  44579  hoiqssbllem2  44586  hspmbllem2  44590  opnvonmbllem2  44596  iinhoiicc  44637  iunhoiioo  44639  vonioo  44645  vonicc  44648
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