![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6481 | . 2 ⊢ ( I ↾ V):V–1-1-onto→V | |
2 | reli 5548 | . . . 4 ⊢ Rel I | |
3 | dfrel3 5894 | . . . 4 ⊢ (Rel I ↔ ( I ↾ V) = I ) | |
4 | 2, 3 | mpbi 222 | . . 3 ⊢ ( I ↾ V) = I |
5 | f1oeq1 6433 | . . 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 222 | 1 ⊢ I :V–1-1-onto→V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 Vcvv 3415 I cid 5311 ↾ cres 5409 Rel wrel 5412 –1-1-onto→wf1o 6187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-br 4930 df-opab 4992 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 |
This theorem is referenced by: ncanth 6935 |
Copyright terms: Public domain | W3C validator |