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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 6645 | . 2 ⊢ ( I ↾ V):V–1-1-onto→V | |
2 | reli 5691 | . . . 4 ⊢ Rel I | |
3 | dfrel3 6048 | . . . 4 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
4 | 2, 3 | mpbi 231 | . . 3 ⊢ ( I ↾ V) = I |
5 | f1oeq1 6597 | . . 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 231 | 1 ⊢ I :V–1-1-onto→V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 Vcvv 3492 I cid 5452 ↾ cres 5550 Rel wrel 5553 –1-1-onto→wf1o 6347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 |
This theorem is referenced by: ncanth 7101 |
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