Theorem nfixpw 8892

Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8893 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2402. (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 8874	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑥
4		nfixpw.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
5	3, 4	nfel 2937	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
6	5	nfab 2929	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
8	7	mptru 1566	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
9	2, 8	nffn 6615	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
10		df-ral 3076	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
11		nftru 1823	. . . . . . 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 6874	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
16		nfixpw.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
17	16	a1i 11	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
18	15, 17	nfeld 2934	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
19	12, 18	nfimd 1913	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
20	11, 19	nfald 2359	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
21	20	mptru 1566	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
22	10, 21	nfxfr 1872	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
23	9, 22	nfan 1918	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
24	23	nfab 2929	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
25	1, 24	nfcxfr 2921	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557  ⊤wtru 1560  Ⅎwnf 1802   ∈ wcel 2141  {cab 2739  Ⅎwnfc 2908  ∀wral 3075   Fn wfn 6511  ‘cfv 6516  Xcixp 8873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fn 6519  df-fv 6524  df-ixp 8874

This theorem is referenced by:  vonioo  47217