Theorem nfixpxy 6929

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 6911	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nftru 1515	. . . . . . 7 ⊢ Ⅎ𝑥⊤
4		nfcvd 2376	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝑥)
5		nfixp.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
6	5	a1i 9	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝐴)
7	4, 6	nfeld 2391	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
8	3, 7	nfabd 2395	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
9	8	mptru 1407	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
10	2, 9	nffn 5433	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
11		df-ral 2516	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
12	2	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧)
13	12, 4	nffvd 5660	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
14		nfixp.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
15	14	a1i 9	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
16	13, 15	nfeld 2391	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
17	7, 16	nfimd 1634	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
18	3, 17	nfald 1808	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
19	18	mptru 1407	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
20	11, 19	nfxfr 1523	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
21	10, 20	nfan 1614	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
22	21	nfab 2380	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
23	1, 22	nfcxfr 2372	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396  ⊤wtru 1399  Ⅎwnf 1509   ∈ wcel 2202  {cab 2217  Ⅎwnfc 2362  ∀wral 2511   Fn wfn 5328  ‘cfv 5333  Xcixp 6910

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ixp 6911

This theorem is referenced by: (None)