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Mirrors > Home > ILE Home > Th. List > f1ovi | GIF version |
Description: The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.) |
Ref | Expression |
---|---|
f1ovi | ⊢ I :V–1-1-onto→V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 5164 | . 2 ⊢ ( I ↾ V):V–1-1-onto→V | |
2 | reli 4465 | . . . 4 ⊢ Rel I | |
3 | dfrel3 4778 | . . . 4 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
4 | 2, 3 | mpbi 133 | . . 3 ⊢ ( I ↾ V) = I |
5 | f1oeq1 5117 | . . 3 ⊢ (( I ↾ V) = I → (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V)) | |
6 | 4, 5 | ax-mp 7 | . 2 ⊢ (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V) |
7 | 1, 6 | mpbi 133 | 1 ⊢ I :V–1-1-onto→V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 Vcvv 2557 I cid 4025 ↾ cres 4347 Rel wrel 4350 –1-1-onto→wf1o 4901 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: (None) |
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