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Theorem nfixp1 8957
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 8937 . 2 X𝑥𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
2 nfcv 2903 . . . . 5 𝑥𝑦
3 nfab1 2905 . . . . 5 𝑥{𝑥𝑥𝐴}
42, 3nffn 6668 . . . 4 𝑥 𝑦 Fn {𝑥𝑥𝐴}
5 nfra1 3282 . . . 4 𝑥𝑥𝐴 (𝑦𝑥) ∈ 𝐵
64, 5nfan 1897 . . 3 𝑥(𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)
76nfab 2909 . 2 𝑥{𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
81, 7nfcxfr 2901 1 𝑥X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2106  {cab 2712  wnfc 2888  wral 3059   Fn wfn 6558  cfv 6563  Xcixp 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-fun 6565  df-fn 6566  df-ixp 8937
This theorem is referenced by:  ixpiunwdom  9628  ptbasfi  23605  hoidmvlelem3  46553  hspdifhsp  46572  hoiqssbllem2  46579  hspmbllem2  46583  opnvonmbllem2  46589  iinhoiicc  46630  iunhoiioo  46632  vonioo  46638  vonicc  46641
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