Theorem nfixpxy 6785

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 6767	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nftru 1480	. . . . . . 7 ⊢ Ⅎ𝑥⊤
4		nfcvd 2340	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝑥)
5		nfixp.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
6	5	a1i 9	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝐴)
7	4, 6	nfeld 2355	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
8	3, 7	nfabd 2359	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
9	8	mptru 1373	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
10	2, 9	nffn 5355	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
11		df-ral 2480	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
12	2	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧)
13	12, 4	nffvd 5573	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
14		nfixp.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
15	14	a1i 9	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
16	13, 15	nfeld 2355	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
17	7, 16	nfimd 1599	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
18	3, 17	nfald 1774	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
19	18	mptru 1373	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
20	11, 19	nfxfr 1488	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
21	10, 20	nfan 1579	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
22	21	nfab 2344	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
23	1, 22	nfcxfr 2336	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  ⊤wtru 1365  Ⅎwnf 1474   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326  ∀wral 2475   Fn wfn 5254  ‘cfv 5259  Xcixp 6766

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ixp 6767

This theorem is referenced by: (None)