Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dff1o6 | Structured version Visualization version Unicode version |
Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
Ref | Expression |
---|---|
dff1o6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 5895 | . 2 | |
2 | dff13 6512 | . . 3 | |
3 | df-fo 5894 | . . 3 | |
4 | 2, 3 | anbi12i 733 | . 2 |
5 | df-3an 1039 | . . 3 | |
6 | eqimss 3657 | . . . . . . 7 | |
7 | 6 | anim2i 593 | . . . . . 6 |
8 | df-f 5892 | . . . . . 6 | |
9 | 7, 8 | sylibr 224 | . . . . 5 |
10 | 9 | pm4.71ri 665 | . . . 4 |
11 | 10 | anbi1i 731 | . . 3 |
12 | an32 839 | . . 3 | |
13 | 5, 11, 12 | 3bitrri 287 | . 2 |
14 | 1, 4, 13 | 3bitri 286 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wral 2912 wss 3574 crn 5115 wfn 5883 wf 5884 wf1 5885 wfo 5886 wf1o 5887 cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: soisores 6577 f1otrg 25751 f1otrge 25752 grpoinvf 27386 bra11 28967 hgt750lemb 30734 diaf11N 36338 dibf11N 36450 lcfrlem9 36839 mapd1o 36937 hdmapf1oN 37157 hgmapf1oN 37195 rmxypairf1o 37476 |
Copyright terms: Public domain | W3C validator |