Theorem ixp0x 6826

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 6800	. 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
2		velsn 3655	. . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3		fn0 5405	. . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4		ral0 3566	. . . . 5 ⊢ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴
5	4	biantru 302	. . . 4 ⊢ (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴))
6	2, 3, 5	3bitr2i 208	. . 3 ⊢ (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴))
7	6	abbi2i 2321	. 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)}
8	1, 7	eqtr4i 2230	1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅}

Syntax hints:   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∅c0 3464  {csn 3638   Fn wfn 5275  ‘cfv 5280  Xcixp 6798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-fun 5282  df-fn 5283  df-ixp 6799

This theorem is referenced by:  0elixp  6829