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	⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfixp1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ixp 8916	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
2		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfab1 2894	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
4	2, 3	nffn 6648	. . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
5		nfra1 3272	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵
6	4, 5	nfan 1895	. . 3 ⊢ Ⅎ𝑥(𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)
7	6	nfab 2898	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
8	1, 7	nfcxfr 2890	1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 394   ∈ wcel 2099  {cab 2703  Ⅎwnfc 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