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Theorem adjbd1o 32063

Description: The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adjbd1o	⊢ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp

Proof of Theorem adjbd1o Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		adj1o 31872	. . . 4 ⊢ adj_ℎ:dom adj_ℎ–1-1-onto→dom adj_ℎ
2		f1of1 6762	. . . 4 ⊢ (adj_ℎ:dom adj_ℎ–1-1-onto→dom adj_ℎ → adj_ℎ:dom adj_ℎ–1-1→dom adj_ℎ)
3	1, 2	ax-mp 5	. . 3 ⊢ adj_ℎ:dom adj_ℎ–1-1→dom adj_ℎ
4		bdopssadj 32059	. . 3 ⊢ BndLinOp ⊆ dom adj_ℎ
5		f1ores 6777	. . 3 ⊢ ((adj_ℎ:dom adj_ℎ–1-1→dom adj_ℎ ∧ BndLinOp ⊆ dom adj_ℎ) → (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adj_ℎ “ BndLinOp))
6	3, 4, 5	mp2an 692	. 2 ⊢ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adj_ℎ “ BndLinOp)
7		vex 3440	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	elima 6014	. . . . 5 ⊢ (𝑦 ∈ (adj_ℎ “ BndLinOp) ↔ ∃𝑥 ∈ BndLinOp 𝑥adj_ℎ𝑦)
9		f1ofn 6764	. . . . . . . 8 ⊢ (adj_ℎ:dom adj_ℎ–1-1-onto→dom adj_ℎ → adj_ℎ Fn dom adj_ℎ)
10	1, 9	ax-mp 5	. . . . . . 7 ⊢ adj_ℎ Fn dom adj_ℎ
11		bdopadj 32060	. . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → 𝑥 ∈ dom adj_ℎ)
12		fnbrfvb 6872	. . . . . . 7 ⊢ ((adj_ℎ Fn dom adj_ℎ ∧ 𝑥 ∈ dom adj_ℎ) → ((adj_ℎ‘𝑥) = 𝑦 ↔ 𝑥adj_ℎ𝑦))
13	10, 11, 12	sylancr 587	. . . . . 6 ⊢ (𝑥 ∈ BndLinOp → ((adj_ℎ‘𝑥) = 𝑦 ↔ 𝑥adj_ℎ𝑦))
14	13	rexbiia 3077	. . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ BndLinOp 𝑥adj_ℎ𝑦)
15		adjbdlnb 32062	. . . . . . . . 9 ⊢ (𝑥 ∈ BndLinOp ↔ (adj_ℎ‘𝑥) ∈ BndLinOp)
16		eleq1 2819	. . . . . . . . 9 ⊢ ((adj_ℎ‘𝑥) = 𝑦 → ((adj_ℎ‘𝑥) ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp))
17	15, 16	bitrid 283	. . . . . . . 8 ⊢ ((adj_ℎ‘𝑥) = 𝑦 → (𝑥 ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp))
18	17	biimpcd 249	. . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → ((adj_ℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp))
19	18	rexlimiv 3126	. . . . . 6 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp)
20		adjbdln 32061	. . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adj_ℎ‘𝑦) ∈ BndLinOp)
21		bdopadj 32060	. . . . . . . 8 ⊢ (𝑦 ∈ BndLinOp → 𝑦 ∈ dom adj_ℎ)
22		adjadj 31914	. . . . . . . 8 ⊢ (𝑦 ∈ dom adj_ℎ → (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦)
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦)
24		fveqeq2 6831	. . . . . . . 8 ⊢ (𝑥 = (adj_ℎ‘𝑦) → ((adj_ℎ‘𝑥) = 𝑦 ↔ (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦))
25	24	rspcev 3577	. . . . . . 7 ⊢ (((adj_ℎ‘𝑦) ∈ BndLinOp ∧ (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦) → ∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦)
26	20, 23, 25	syl2anc 584	. . . . . 6 ⊢ (𝑦 ∈ BndLinOp → ∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦)
27	19, 26	impbii 209	. . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 ↔ 𝑦 ∈ BndLinOp)
28	8, 14, 27	3bitr2i 299	. . . 4 ⊢ (𝑦 ∈ (adj_ℎ “ BndLinOp) ↔ 𝑦 ∈ BndLinOp)
29	28	eqriv 2728	. . 3 ⊢ (adj_ℎ “ BndLinOp) = BndLinOp
30		f1oeq3 6753	. . 3 ⊢ ((adj_ℎ “ BndLinOp) = BndLinOp → ((adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adj_ℎ “ BndLinOp) ↔ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp))
31	29, 30	ax-mp 5	. 2 ⊢ ((adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adj_ℎ “ BndLinOp) ↔ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp)
32	6, 31	mpbi 230	1 ⊢ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 ⊆ wss 3902 class class class wbr 5091 dom cdm 5616 ↾ cres 5618 “ cima 5619 Fn wfn 6476 –1-1→wf1 6478 –1-1-onto→wf1o 6480 ‘cfv 6481 BndLinOpcbo 30926 adj_ℎcado 30933

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cc 10326 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086 ax-hilex 30977 ax-hfvadd 30978 ax-hvcom 30979 ax-hvass 30980 ax-hv0cl 30981 ax-hvaddid 30982 ax-hfvmul 30983 ax-hvmulid 30984 ax-hvmulass 30985 ax-hvdistr1 30986 ax-hvdistr2 30987 ax-hvmul0 30988 ax-hfi 31057 ax-his1 31060 ax-his2 31061 ax-his3 31062 ax-his4 31063 ax-hcompl 31180

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19230 df-cmn 19695 df-psmet 21284 df-xmet 21285 df-met 21286 df-bl 21287 df-mopn 21288 df-fbas 21289 df-fg 21290 df-cnfld 21293 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-cn 23143 df-cnp 23144 df-lm 23145 df-t1 23230 df-haus 23231 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cfil 25183 df-cau 25184 df-cmet 25185 df-grpo 30471 df-gid 30472 df-ginv 30473 df-gdiv 30474 df-ablo 30523 df-vc 30537 df-nv 30570 df-va 30573 df-ba 30574 df-sm 30575 df-0v 30576 df-vs 30577 df-nmcv 30578 df-ims 30579 df-dip 30679 df-ssp 30700 df-ph 30791 df-cbn 30841 df-hnorm 30946 df-hba 30947 df-hvsub 30949 df-hlim 30950 df-hcau 30951 df-sh 31185 df-ch 31199 df-oc 31230 df-ch0 31231 df-shs 31286 df-pjh 31373 df-h0op 31726 df-nmop 31817 df-cnop 31818 df-lnop 31819 df-bdop 31820 df-unop 31821 df-hmop 31822 df-nmfn 31823 df-nlfn 31824 df-cnfn 31825 df-lnfn 31826 df-adjh 31827

This theorem is referenced by: (None)

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