Theorem nfixp1 8848

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 8828	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
2		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfab1 2897	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴}
4	2, 3	nffn 6585	. . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
5		nfra1 3257	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵
6	4, 5	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)
7	6	nfab 2901	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)}
8	1, 7	nfcxfr 2893	1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∈ wcel 2113  {cab 2711  Ⅎwnfc 2880  ∀wral 3048   Fn wfn 6481  ‘cfv 6486  Xcixp 8827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-fun 6488  df-fn 6489  df-ixp 8828

This theorem is referenced by:  ixpiunwdom  9483  ptbasfi  23497  hoidmvlelem3  46719  hspdifhsp  46738  hoiqssbllem2  46745  hspmbllem2  46749  opnvonmbllem2  46755  iinhoiicc  46796  iunhoiioo  46798  vonioo  46804  vonicc  46807