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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 31709 | . . 3 ⊢ Fun adjℎ | |
2 | funfn 6583 | . . 3 ⊢ (Fun adjℎ ↔ adjℎ Fn dom adjℎ) | |
3 | 1, 2 | mpbi 229 | . 2 ⊢ adjℎ Fn dom adjℎ |
4 | funcnvadj 31716 | . 2 ⊢ Fun ◡adjℎ | |
5 | df-rn 5689 | . . 3 ⊢ ran adjℎ = dom ◡adjℎ | |
6 | cnvadj 31715 | . . . 4 ⊢ ◡adjℎ = adjℎ | |
7 | 6 | dmeqi 5907 | . . 3 ⊢ dom ◡adjℎ = dom adjℎ |
8 | 5, 7 | eqtri 2756 | . 2 ⊢ ran adjℎ = dom adjℎ |
9 | dff1o2 6844 | . 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 1339 | 1 ⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ◡ccnv 5677 dom cdm 5678 ran crn 5679 Fun wfun 6542 Fn wfn 6543 –1-1-onto→wf1o 6547 adjℎcado 30778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-hfvadd 30823 ax-hvcom 30824 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvdistr2 30832 ax-hvmul0 30833 ax-hfi 30902 ax-his1 30905 ax-his2 30906 ax-his3 30907 ax-his4 30908 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-2 12306 df-cj 15079 df-re 15080 df-im 15081 df-hvsub 30794 df-adjh 31672 |
This theorem is referenced by: dmadjrn 31718 adjbdlnb 31907 adjbd1o 31908 |
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