Theorem ixp0x 8940

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 8913	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
2		velsn 4617	. . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3		fn0 6669	. . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4		ral0 4488	. . . . 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 2870	. 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
8	1, 7	eqtr4i 2761	1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∅c0 4308  {csn 4601   Fn wfn 6526  ‘cfv 6531  Xcixp 8911

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-fun 6533  df-fn 6534  df-ixp 8912

This theorem is referenced by:  0elixp  8943  ptcmpfi  23751  finixpnum  37629  ioorrnopn  46334  ioorrnopnxr  46336  hoicvr  46577  ovnhoi  46632  ovnlecvr2  46639  hoiqssbl  46654  hoimbl  46660  iunhoiioo  46705