Theorem nfixpw 8753

Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8754 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 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 8735	. 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 1547	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
9	2, 8	nffn 6570	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
10		df-ral 3062	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
11		nftru 1805	. . . . . . 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 6823	. . . . . . . . 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 1896	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
20	11, 19	nfald 2321	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
21	20	mptru 1547	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
22	10, 21	nfxfr 1854	. . . 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 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 396  ∀wal 1538  ⊤wtru 1541  Ⅎwnf 1784   ∈ wcel 2105  {cab 2713  Ⅎwnfc 2884  ∀wral 3061   Fn wfn 6460  ‘cfv 6465  Xcixp 8734

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-opab 5149  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-iota 6417  df-fun 6467  df-fn 6468  df-fv 6473  df-ixp 8735

This theorem is referenced by:  vonioo  44476