Theorem nfixpxy 6864

Description: Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)

Hypotheses

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

Assertion

Ref	Expression
nfixpxy	⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfixpxy

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ixp 6846	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nftru 1512	. . . . . . 7 ⊢ Ⅎ𝑥⊤
4		nfcvd 2373	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝑥)
5		nfixp.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
6	5	a1i 9	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝐴)
7	4, 6	nfeld 2388	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
8	3, 7	nfabd 2392	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
9	8	mptru 1404	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
10	2, 9	nffn 5417	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
11		df-ral 2513	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
12	2	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧)
13	12, 4	nffvd 5639	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
14		nfixp.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
15	14	a1i 9	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
16	13, 15	nfeld 2388	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
17	7, 16	nfimd 1631	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
18	3, 17	nfald 1806	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
19	18	mptru 1404	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
20	11, 19	nfxfr 1520	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
21	10, 20	nfan 1611	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
22	21	nfab 2377	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
23	1, 22	nfcxfr 2369	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1393  ⊤wtru 1396  Ⅎwnf 1506   ∈ wcel 2200  {cab 2215  Ⅎwnfc 2359  ∀wral 2508   Fn wfn 5313  ‘cfv 5318  Xcixp 6845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ixp 6846

This theorem is referenced by: (None)