Theorem nfixpw 8866

Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8867 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2377. (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 8848	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑥
4		nfixpw.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
5	3, 4	nfel 2914	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
6	5	nfab 2905	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
8	7	mptru 1549	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
9	2, 8	nffn 6599	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
10		df-ral 3053	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
11		nftru 1806	. . . . . . 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 6854	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
16		nfixpw.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
17	16	a1i 11	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
18	15, 17	nfeld 2911	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
19	12, 18	nfimd 1896	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
20	11, 19	nfald 2334	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
21	20	mptru 1549	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
22	10, 21	nfxfr 1855	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
23	9, 22	nfan 1901	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
24	23	nfab 2905	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
25	1, 24	nfcxfr 2897	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  ⊤wtru 1543  Ⅎwnf 1785   ∈ wcel 2114  {cab 2715  Ⅎwnfc 2884  ∀wral 3052   Fn wfn 6495  ‘cfv 6500  Xcixp 8847

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-rex 3063  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-uni 4866  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-ixp 8848

This theorem is referenced by:  vonioo  47037