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Theorem nfixp1 8161
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 8142 . 2 X𝑥𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
2 nfcv 2948 . . . . 5 𝑥𝑦
3 nfab1 2950 . . . . 5 𝑥{𝑥𝑥𝐴}
42, 3nffn 6194 . . . 4 𝑥 𝑦 Fn {𝑥𝑥𝐴}
5 nfra1 3129 . . . 4 𝑥𝑥𝐴 (𝑦𝑥) ∈ 𝐵
64, 5nfan 1990 . . 3 𝑥(𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)
76nfab 2953 . 2 𝑥{𝑦 ∣ (𝑦 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑦𝑥) ∈ 𝐵)}
81, 7nfcxfr 2946 1 𝑥X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 2156  {cab 2792  wnfc 2935  wral 3096   Fn wfn 6092  cfv 6097  Xcixp 8141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-fun 6099  df-fn 6100  df-ixp 8142
This theorem is referenced by:  ixpiunwdom  8731  ptbasfi  21594  hoidmvlelem3  41287  hspdifhsp  41306  hoiqssbllem2  41313  hspmbllem2  41317  opnvonmbllem2  41323  iinhoiicc  41364  iunhoiioo  41366  vonioo  41372  vonicc  41375
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