adjbd1o - Hilbert Space Explorer

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Theorem adjbd1o 31915

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 31724	. . . 4 ⊢ adj_ℎ:dom adj_ℎ–1-1-onto→dom adj_ℎ
2		f1of1 6843	. . . 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 31911	. . 3 ⊢ BndLinOp ⊆ dom adj_ℎ
5		f1ores 6858	. . 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 690	. 2 ⊢ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adj_ℎ “ BndLinOp)
7		vex 3477	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	elima 6073	. . . . 5 ⊢ (𝑦 ∈ (adj_ℎ “ BndLinOp) ↔ ∃𝑥 ∈ BndLinOp 𝑥adj_ℎ𝑦)
9		f1ofn 6845	. . . . . . . 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 31912	. . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → 𝑥 ∈ dom adj_ℎ)
12		fnbrfvb 6955	. . . . . . 7 ⊢ ((adj_ℎ Fn dom adj_ℎ ∧ 𝑥 ∈ dom adj_ℎ) → ((adj_ℎ‘𝑥) = 𝑦 ↔ 𝑥adj_ℎ𝑦))
13	10, 11, 12	sylancr 585	. . . . . 6 ⊢ (𝑥 ∈ BndLinOp → ((adj_ℎ‘𝑥) = 𝑦 ↔ 𝑥adj_ℎ𝑦))
14	13	rexbiia 3089	. . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ BndLinOp 𝑥adj_ℎ𝑦)
15		adjbdlnb 31914	. . . . . . . . 9 ⊢ (𝑥 ∈ BndLinOp ↔ (adj_ℎ‘𝑥) ∈ BndLinOp)
16		eleq1 2817	. . . . . . . . 9 ⊢ ((adj_ℎ‘𝑥) = 𝑦 → ((adj_ℎ‘𝑥) ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp))
17	15, 16	bitrid 282	. . . . . . . 8 ⊢ ((adj_ℎ‘𝑥) = 𝑦 → (𝑥 ∈ BndLinOp ↔ 𝑦 ∈ BndLinOp))
18	17	biimpcd 248	. . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → ((adj_ℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp))
19	18	rexlimiv 3145	. . . . . 6 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp)
20		adjbdln 31913	. . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adj_ℎ‘𝑦) ∈ BndLinOp)
21		bdopadj 31912	. . . . . . . 8 ⊢ (𝑦 ∈ BndLinOp → 𝑦 ∈ dom adj_ℎ)
22		adjadj 31766	. . . . . . . 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 6911	. . . . . . . 8 ⊢ (𝑥 = (adj_ℎ‘𝑦) → ((adj_ℎ‘𝑥) = 𝑦 ↔ (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦))
25	24	rspcev 3611	. . . . . . 7 ⊢ (((adj_ℎ‘𝑦) ∈ BndLinOp ∧ (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦) → ∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦)
26	20, 23, 25	syl2anc 582	. . . . . 6 ⊢ (𝑦 ∈ BndLinOp → ∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦)
27	19, 26	impbii 208	. . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 ↔ 𝑦 ∈ BndLinOp)
28	8, 14, 27	3bitr2i 298	. . . 4 ⊢ (𝑦 ∈ (adj_ℎ “ BndLinOp) ↔ 𝑦 ∈ BndLinOp)
29	28	eqriv 2725	. . 3 ⊢ (adj_ℎ “ BndLinOp) = BndLinOp
30		f1oeq3 6834	. . 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 229	1 ⊢ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 ⊆ wss 3949 class class class wbr 5152 dom cdm 5682 ↾ cres 5684 “ cima 5685 Fn wfn 6548 –1-1→wf1 6550 –1-1-onto→wf1o 6552 ‘cfv 6553 BndLinOpcbo 30778 adj_ℎcado 30785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cc 10466 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 ax-hilex 30829 ax-hfvadd 30830 ax-hvcom 30831 ax-hvass 30832 ax-hv0cl 30833 ax-hvaddid 30834 ax-hfvmul 30835 ax-hvmulid 30836 ax-hvmulass 30837 ax-hvdistr1 30838 ax-hvdistr2 30839 ax-hvmul0 30840 ax-hfi 30909 ax-his1 30912 ax-his2 30913 ax-his3 30914 ax-his4 30915 ax-hcompl 31032

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-cn 23151 df-cnp 23152 df-lm 23153 df-t1 23238 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cfil 25203 df-cau 25204 df-cmet 25205 df-grpo 30323 df-gid 30324 df-ginv 30325 df-gdiv 30326 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-vs 30429 df-nmcv 30430 df-ims 30431 df-dip 30531 df-ssp 30552 df-ph 30643 df-cbn 30693 df-hnorm 30798 df-hba 30799 df-hvsub 30801 df-hlim 30802 df-hcau 30803 df-sh 31037 df-ch 31051 df-oc 31082 df-ch0 31083 df-shs 31138 df-pjh 31225 df-h0op 31578 df-nmop 31669 df-cnop 31670 df-lnop 31671 df-bdop 31672 df-unop 31673 df-hmop 31674 df-nmfn 31675 df-nlfn 31676 df-cnfn 31677 df-lnfn 31678 df-adjh 31679

This theorem is referenced by: (None)

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