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Theorem adjbd1o 32172

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 31981	. . . 4 ⊢ adj_ℎ:dom adj_ℎ–1-1-onto→dom adj_ℎ
2		f1of1 6781	. . . 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 32168	. . 3 ⊢ BndLinOp ⊆ dom adj_ℎ
5		f1ores 6796	. . 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 693	. 2 ⊢ (adj_ℎ ↾ BndLinOp):BndLinOp–1-1-onto→(adj_ℎ “ BndLinOp)
7		vex 3446	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	elima 6032	. . . . 5 ⊢ (𝑦 ∈ (adj_ℎ “ BndLinOp) ↔ ∃𝑥 ∈ BndLinOp 𝑥adj_ℎ𝑦)
9		f1ofn 6783	. . . . . . . 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 32169	. . . . . . 7 ⊢ (𝑥 ∈ BndLinOp → 𝑥 ∈ dom adj_ℎ)
12		fnbrfvb 6892	. . . . . . 7 ⊢ ((adj_ℎ Fn dom adj_ℎ ∧ 𝑥 ∈ dom adj_ℎ) → ((adj_ℎ‘𝑥) = 𝑦 ↔ 𝑥adj_ℎ𝑦))
13	10, 11, 12	sylancr 588	. . . . . 6 ⊢ (𝑥 ∈ BndLinOp → ((adj_ℎ‘𝑥) = 𝑦 ↔ 𝑥adj_ℎ𝑦))
14	13	rexbiia 3083	. . . . 5 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ BndLinOp 𝑥adj_ℎ𝑦)
15		adjbdlnb 32171	. . . . . . . . 9 ⊢ (𝑥 ∈ BndLinOp ↔ (adj_ℎ‘𝑥) ∈ BndLinOp)
16		eleq1 2825	. . . . . . . . 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 3132	. . . . . 6 ⊢ (∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦 → 𝑦 ∈ BndLinOp)
20		adjbdln 32170	. . . . . . 7 ⊢ (𝑦 ∈ BndLinOp → (adj_ℎ‘𝑦) ∈ BndLinOp)
21		bdopadj 32169	. . . . . . . 8 ⊢ (𝑦 ∈ BndLinOp → 𝑦 ∈ dom adj_ℎ)
22		adjadj 32023	. . . . . . . 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 6851	. . . . . . . 8 ⊢ (𝑥 = (adj_ℎ‘𝑦) → ((adj_ℎ‘𝑥) = 𝑦 ↔ (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦))
25	24	rspcev 3578	. . . . . . 7 ⊢ (((adj_ℎ‘𝑦) ∈ BndLinOp ∧ (adj_ℎ‘(adj_ℎ‘𝑦)) = 𝑦) → ∃𝑥 ∈ BndLinOp (adj_ℎ‘𝑥) = 𝑦)
26	20, 23, 25	syl2anc 585	. . . . . 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 2734	. . 3 ⊢ (adj_ℎ “ BndLinOp) = BndLinOp
30		f1oeq3 6772	. . 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 1542 ∈ wcel 2114 ∃wrex 3062 ⊆ wss 3903 class class class wbr 5100 dom cdm 5632 ↾ cres 5634 “ cima 5635 Fn wfn 6495 –1-1→wf1 6497 –1-1-onto→wf1o 6499 ‘cfv 6500 BndLinOpcbo 31035 adj_ℎcado 31042

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31086 ax-hfvadd 31087 ax-hvcom 31088 ax-hvass 31089 ax-hv0cl 31090 ax-hvaddid 31091 ax-hfvmul 31092 ax-hvmulid 31093 ax-hvmulass 31094 ax-hvdistr1 31095 ax-hvdistr2 31096 ax-hvmul0 31097 ax-hfi 31166 ax-his1 31169 ax-his2 31170 ax-his3 31171 ax-his4 31172 ax-hcompl 31289

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-cn 23183 df-cnp 23184 df-lm 23185 df-t1 23270 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cfil 25223 df-cau 25224 df-cmet 25225 df-grpo 30580 df-gid 30581 df-ginv 30582 df-gdiv 30583 df-ablo 30632 df-vc 30646 df-nv 30679 df-va 30682 df-ba 30683 df-sm 30684 df-0v 30685 df-vs 30686 df-nmcv 30687 df-ims 30688 df-dip 30788 df-ssp 30809 df-ph 30900 df-cbn 30950 df-hnorm 31055 df-hba 31056 df-hvsub 31058 df-hlim 31059 df-hcau 31060 df-sh 31294 df-ch 31308 df-oc 31339 df-ch0 31340 df-shs 31395 df-pjh 31482 df-h0op 31835 df-nmop 31926 df-cnop 31927 df-lnop 31928 df-bdop 31929 df-unop 31930 df-hmop 31931 df-nmfn 31932 df-nlfn 31933 df-cnfn 31934 df-lnfn 31935 df-adjh 31936

This theorem is referenced by: (None)

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