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Mirrors > Home > MPE Home > Th. List > f1ovi | Structured version Visualization version 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 6335 | . 2 ⊢ ( I ↾ V):V–1-1-onto→V | |
2 | reli 5405 | . . . 4 ⊢ Rel I | |
3 | dfrel3 5750 | . . . 4 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
4 | 2, 3 | mpbi 220 | . . 3 ⊢ ( I ↾ V) = I |
5 | f1oeq1 6288 | . . 3 ⊢ (( I ↾ V) = I → (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V) |
7 | 1, 6 | mpbi 220 | 1 ⊢ I :V–1-1-onto→V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1632 Vcvv 3340 I cid 5173 ↾ cres 5268 Rel wrel 5271 –1-1-onto→wf1o 6048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 |
This theorem is referenced by: ncanth 6772 |
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