Theorem ixp0x 8947

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 8920	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
2		velsn 4600	. . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3		fn0 6668	. . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4		ral0 4454	. . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴
5	4	biantru 539	. . . 4 ⊢ (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴))
6	2, 3, 5	3bitr2i 302	. . 3 ⊢ (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴))
7	6	eqabi 2896	. 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
8	1, 7	eqtr4i 2787	1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∅c0 4279  {csn 4584   Fn wfn 6532  ‘cfv 6537  Xcixp 8918

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-fun 6539  df-fn 6540  df-ixp 8919

This theorem is used by:  0elixp  8950  ptcmpfi  24125  finixpnum  38508  ioorrnopn  47284  ioorrnopnxr  47286  hoicvr  47527  ovnhoi  47582  ovnlecvr2  47589  hoiqssbl  47604  hoimbl  47610  iunhoiioo  47655