Theorem nfixp 8915

Description: Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker nfixpw 8914 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfixp	⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfixp

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ixp 8896	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
2		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑦𝑧
3		nftru 1804	. . . . . . 7 ⊢ Ⅎ𝑥⊤
4		nfcvf 2930	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
5	4	adantl 480	. . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝑥)
6		nfixp.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
7	6	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝐴)
8	5, 7	nfeld 2912	. . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥 ∈ 𝐴)
9	3, 8	nfabd2 2927	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴})
10	9	mptru 1546	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}
11	2, 10	nffn 6649	. . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴}
12		df-ral 3060	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
13	2	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝑧)
14	13, 5	nffvd 6904	. . . . . . . . 9 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧‘𝑥))
15		nfixp.2	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
16	15	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝐵)
17	14, 16	nfeld 2912	. . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵)
18	8, 17	nfimd 1895	. . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
19	3, 18	nfald2 2442	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
20	19	mptru 1546	. . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)
21	12, 20	nfxfr 1853	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵
22	11, 21	nfan 1900	. . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)
23	22	nfab 2907	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
24	1, 23	nfcxfr 2899	1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394  ∀wal 1537  ⊤wtru 1540  Ⅎwnf 1783   ∈ wcel 2104  {cab 2707  Ⅎwnfc 2881  ∀wral 3059   Fn wfn 6539  ‘cfv 6544  Xcixp 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-13 2369  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-ixp 8896

This theorem is referenced by: (None)