Theorem nfixpw 8889

Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8890 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)

Hypotheses

Ref	Expression
nfixpw.1	⊢ Ⅎ𝑦𝐴
nfixpw.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfixpw	⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfixpw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ixp 8871	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑥
4		nfixpw.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
5	3, 4	nfel 2906	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
6	5	nfab 2897	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
8	7	mptru 1547	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
9	2, 8	nffn 6617	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
10		df-ral 3045	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
11		nftru 1804	. . . . . . 7 ⊢ Ⅎ𝑥⊤
12	5	a1i 11	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
13	2	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧)
14	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑥)
15	13, 14	nffvd 6870	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
16		nfixpw.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
17	16	a1i 11	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
18	15, 17	nfeld 2903	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
19	12, 18	nfimd 1894	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
20	11, 19	nfald 2327	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
21	20	mptru 1547	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
22	10, 21	nfxfr 1853	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
23	9, 22	nfan 1899	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
24	23	nfab 2897	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
25	1, 24	nfcxfr 2889	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  ⊤wtru 1541  Ⅎwnf 1783   ∈ wcel 2109  {cab 2707  Ⅎwnfc 2876  ∀wral 3044   Fn wfn 6506  ‘cfv 6511  Xcixp 8870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519  df-ixp 8871

This theorem is referenced by:  vonioo  46680