Theorem nfixpw 8597

Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8598 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 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 8579	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑥
4		nfixpw.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
5	3, 4	nfel 2918	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
6	5	nfab 2910	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
8	7	mptru 1550	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
9	2, 8	nffn 6478	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
10		df-ral 3066	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
11		nftru 1812	. . . . . . 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 6729	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
16		nfixpw.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
17	16	a1i 11	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
18	15, 17	nfeld 2915	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
19	12, 18	nfimd 1902	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
20	11, 19	nfald 2327	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
21	20	mptru 1550	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
22	10, 21	nfxfr 1860	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
23	9, 22	nfan 1907	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
24	23	nfab 2910	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
25	1, 24	nfcxfr 2902	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541  ⊤wtru 1544  Ⅎwnf 1791   ∈ wcel 2110  {cab 2714  Ⅎwnfc 2884  ∀wral 3061   Fn wfn 6375  ‘cfv 6380  Xcixp 8578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-ixp 8579

This theorem is referenced by:  vonioo  43895