Theorem ixp0x 8876

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 8849	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
2		velsn 4598	. . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3		fn0 6631	. . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4		ral0 4453	. . . . 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 2872	. 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
8	1, 7	eqtr4i 2763	1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∅c0 4287  {csn 4582   Fn wfn 6495  ‘cfv 6500  Xcixp 8847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-fun 6502  df-fn 6503  df-ixp 8848

This theorem is referenced by:  0elixp  8879  ptcmpfi  23769  finixpnum  37856  ioorrnopn  46663  ioorrnopnxr  46665  hoicvr  46906  ovnhoi  46961  ovnlecvr2  46968  hoiqssbl  46983  hoimbl  46989  iunhoiioo  47034