Theorem ixp0x 8917

Description: An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)

Assertion

Ref	Expression
ixp0x	⊢ X𝑥 ∈ ∅ 𝐴 = {∅}

Proof of Theorem ixp0x

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfixp 8890	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
2		velsn 4637	. . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3		fn0 6672	. . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4		ral0 4505	. . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴
5	4	biantru 529	. . . 4 ⊢ (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴))
6	2, 3, 5	3bitr2i 299	. . 3 ⊢ (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴))
7	6	eqabi 2861	. 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
8	1, 7	eqtr4i 2755	1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  ∀wral 3053  ∅c0 4315  {csn 4621   Fn wfn 6529  ‘cfv 6534  Xcixp 8888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-fun 6536  df-fn 6537  df-ixp 8889

This theorem is referenced by:  0elixp  8920  ptcmpfi  23661  finixpnum  36976  ioorrnopn  45566  ioorrnopnxr  45568  hoicvr  45809  ovnhoi  45864  ovnlecvr2  45871  hoiqssbl  45886  hoimbl  45892  iunhoiioo  45937