Theorem nfixpxy 6611

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 6593	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nftru 1442	. . . . . . 7 ⊢ Ⅎ𝑥⊤
4		nfcvd 2282	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝑥)
5		nfixp.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
6	5	a1i 9	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝐴)
7	4, 6	nfeld 2297	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
8	3, 7	nfabd 2300	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
9	8	mptru 1340	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
10	2, 9	nffn 5219	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
11		df-ral 2421	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
12	2	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧)
13	12, 4	nffvd 5433	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥))
14		nfixp.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
15	14	a1i 9	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵)
16	13, 15	nfeld 2297	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
17	7, 16	nfimd 1564	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
18	3, 17	nfald 1733	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
19	18	mptru 1340	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
20	11, 19	nfxfr 1450	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
21	10, 20	nfan 1544	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
22	21	nfab 2286	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
23	1, 22	nfcxfr 2278	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329  ⊤wtru 1332  Ⅎwnf 1436   ∈ wcel 1480  {cab 2125  Ⅎwnfc 2268  ∀wral 2416   Fn wfn 5118  ‘cfv 5123  Xcixp 6592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-ixp 6593

This theorem is referenced by: (None)