Theorem nfixp1 8959

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	⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfixp1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ixp 8939	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
2		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfab1 2906	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
4	2, 3	nffn 6666	. . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
5		nfra1 3283	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵
6	4, 5	nfan 1898	. . 3 ⊢ Ⅎ𝑥(𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)
7	6	nfab 2910	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
8	1, 7	nfcxfr 2902	1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∈ wcel 2107  {cab 2713  Ⅎwnfc 2889  ∀wral 3060   Fn wfn 6555  ‘cfv 6560  Xcixp 8938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-fun 6562  df-fn 6563  df-ixp 8939

This theorem is referenced by:  ixpiunwdom  9631  ptbasfi  23590  hoidmvlelem3  46617  hspdifhsp  46636  hoiqssbllem2  46643  hspmbllem2  46647  opnvonmbllem2  46653  iinhoiicc  46694  iunhoiioo  46696  vonioo  46702  vonicc  46705