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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 5107 | . 2 ⊢ ( I ↾ V):V–1-1-onto→V | |
2 | reli 4408 | . . . 4 ⊢ Rel I | |
3 | dfrel3 4721 | . . . 4 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
4 | 2, 3 | mpbi 133 | . . 3 ⊢ ( I ↾ V) = I |
5 | f1oeq1 5060 | . . 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 1242 Vcvv 2551 I cid 4016 ↾ cres 4290 Rel wrel 4293 –1-1-onto→wf1o 4844 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 |
This theorem is referenced by: (None) |
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