Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > f0 | GIF version |
Description: The empty function. (Contributed by NM, 14-Aug-1999.) |
Ref | Expression |
---|---|
f0 | ⊢ ∅:∅⟶𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2117 | . . 3 ⊢ ∅ = ∅ | |
2 | fn0 5212 | . . 3 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
3 | 1, 2 | mpbir 145 | . 2 ⊢ ∅ Fn ∅ |
4 | rn0 4765 | . . 3 ⊢ ran ∅ = ∅ | |
5 | 0ss 3371 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 3099 | . 2 ⊢ ran ∅ ⊆ 𝐴 |
7 | df-f 5097 | . 2 ⊢ (∅:∅⟶𝐴 ↔ (∅ Fn ∅ ∧ ran ∅ ⊆ 𝐴)) | |
8 | 3, 6, 7 | mpbir2an 911 | 1 ⊢ ∅:∅⟶𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ⊆ wss 3041 ∅c0 3333 ran crn 4510 Fn wfn 5088 ⟶wf 5089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 |
This theorem is referenced by: f00 5284 f0bi 5285 f10 5369 map0g 6550 ac6sfi 6760 0met 12480 |
Copyright terms: Public domain | W3C validator |