Theorem nfixp1 8870

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 8850	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
2		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfab1 2901	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
4	2, 3	nffn 6601	. . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
5		nfra1 3262	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵
6	4, 5	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)
7	6	nfab 2905	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
8	1, 7	nfcxfr 2897	1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∈ wcel 2114  {cab 2715  Ⅎwnfc 2884  ∀wral 3052   Fn wfn 6497  ‘cfv 6502  Xcixp 8849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6504  df-fn 6505  df-ixp 8850

This theorem is referenced by:  ixpiunwdom  9509  ptbasfi  23542  hoidmvlelem3  46984  hspdifhsp  47003  hoiqssbllem2  47010  hspmbllem2  47014  opnvonmbllem2  47020  iinhoiicc  47061  iunhoiioo  47063  vonioo  47069  vonicc  47072