Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > f1oi | GIF version |
Description: A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
f1oi | ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresi 5235 | . 2 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
2 | cnvresid 5192 | . . . 4 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
3 | 2 | fneq1i 5212 | . . 3 ⊢ (◡( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴) |
4 | 1, 3 | mpbir 145 | . 2 ⊢ ◡( I ↾ 𝐴) Fn 𝐴 |
5 | dff1o4 5368 | . 2 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ◡( I ↾ 𝐴) Fn 𝐴)) | |
6 | 1, 4, 5 | mpbir2an 926 | 1 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 |
Colors of variables: wff set class |
Syntax hints: I cid 4205 ◡ccnv 4533 ↾ cres 4536 Fn wfn 5113 –1-1-onto→wf1o 5117 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 |
This theorem is referenced by: f1ovi 5399 isoid 5704 enrefg 6651 ssdomg 6665 omp1eomlem 6972 ctm 6987 omct 6995 ctssexmid 7017 ssidcn 12368 dvid 12820 dvexp 12833 subctctexmid 13185 |
Copyright terms: Public domain | W3C validator |