| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ixp0x | Structured version Visualization version GIF version | ||
| Description: An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.) |
| Ref | Expression |
|---|---|
| ixp0x | ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfixp 8939 | . 2 ⊢ X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} | |
| 2 | velsn 4642 | . . . 4 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) | |
| 3 | fn0 6699 | . . . 4 ⊢ (𝑓 Fn ∅ ↔ 𝑓 = ∅) | |
| 4 | ral0 4513 | . . . . 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 2877 | . 2 ⊢ {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓‘𝑥) ∈ 𝐴)} |
| 8 | 1, 7 | eqtr4i 2768 | 1 ⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∅c0 4333 {csn 4626 Fn wfn 6556 ‘cfv 6561 Xcixp 8937 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 df-ixp 8938 |
| This theorem is referenced by: 0elixp 8969 ptcmpfi 23821 finixpnum 37612 ioorrnopn 46320 ioorrnopnxr 46322 hoicvr 46563 ovnhoi 46618 ovnlecvr2 46625 hoiqssbl 46640 hoimbl 46646 iunhoiioo 46691 |
| Copyright terms: Public domain | W3C validator |