Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > adj1o | Structured version Visualization version GIF version |
Description: The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adj1o | ⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funadj 29665 | . . 3 ⊢ Fun adjℎ | |
2 | funfn 6387 | . . 3 ⊢ (Fun adjℎ ↔ adjℎ Fn dom adjℎ) | |
3 | 1, 2 | mpbi 232 | . 2 ⊢ adjℎ Fn dom adjℎ |
4 | funcnvadj 29672 | . 2 ⊢ Fun ◡adjℎ | |
5 | df-rn 5568 | . . 3 ⊢ ran adjℎ = dom ◡adjℎ | |
6 | cnvadj 29671 | . . . 4 ⊢ ◡adjℎ = adjℎ | |
7 | 6 | dmeqi 5775 | . . 3 ⊢ dom ◡adjℎ = dom adjℎ |
8 | 5, 7 | eqtri 2846 | . 2 ⊢ ran adjℎ = dom adjℎ |
9 | dff1o2 6622 | . 2 ⊢ (adjℎ:dom adjℎ–1-1-onto→dom adjℎ ↔ (adjℎ Fn dom adjℎ ∧ Fun ◡adjℎ ∧ ran adjℎ = dom adjℎ)) | |
10 | 3, 4, 8, 9 | mpbir3an 1337 | 1 ⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ◡ccnv 5556 dom cdm 5557 ran crn 5558 Fun wfun 6351 Fn wfn 6352 –1-1-onto→wf1o 6356 adjℎcado 28734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-hvsub 28750 df-adjh 29628 |
This theorem is referenced by: dmadjrn 29674 adjbdlnb 29863 adjbd1o 29864 |
Copyright terms: Public domain | W3C validator |