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Mirrors > Home > HSE Home > Th. List > pjrni | Structured version Visualization version GIF version |
Description: The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjfn.1 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjrni | ⊢ ran (projℎ‘𝐻) = 𝐻 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjfn.1 | . . . . 5 ⊢ 𝐻 ∈ Cℋ | |
2 | 1 | pjfni 29253 | . . . 4 ⊢ (projℎ‘𝐻) Fn ℋ |
3 | 1 | pjcli 28969 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐻)‘𝑥) ∈ 𝐻) |
4 | 3 | rgen 3095 | . . . 4 ⊢ ∀𝑥 ∈ ℋ ((projℎ‘𝐻)‘𝑥) ∈ 𝐻 |
5 | ffnfv 6703 | . . . 4 ⊢ ((projℎ‘𝐻): ℋ⟶𝐻 ↔ ((projℎ‘𝐻) Fn ℋ ∧ ∀𝑥 ∈ ℋ ((projℎ‘𝐻)‘𝑥) ∈ 𝐻)) | |
6 | 2, 4, 5 | mpbir2an 698 | . . 3 ⊢ (projℎ‘𝐻): ℋ⟶𝐻 |
7 | frn 6348 | . . 3 ⊢ ((projℎ‘𝐻): ℋ⟶𝐻 → ran (projℎ‘𝐻) ⊆ 𝐻) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ ran (projℎ‘𝐻) ⊆ 𝐻 |
9 | pjid 29247 | . . . . 5 ⊢ ((𝐻 ∈ Cℋ ∧ 𝑦 ∈ 𝐻) → ((projℎ‘𝐻)‘𝑦) = 𝑦) | |
10 | 1, 9 | mpan 677 | . . . 4 ⊢ (𝑦 ∈ 𝐻 → ((projℎ‘𝐻)‘𝑦) = 𝑦) |
11 | 1 | cheli 28782 | . . . . 5 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ) |
12 | fnfvelrn 6671 | . . . . 5 ⊢ (((projℎ‘𝐻) Fn ℋ ∧ 𝑦 ∈ ℋ) → ((projℎ‘𝐻)‘𝑦) ∈ ran (projℎ‘𝐻)) | |
13 | 2, 11, 12 | sylancr 578 | . . . 4 ⊢ (𝑦 ∈ 𝐻 → ((projℎ‘𝐻)‘𝑦) ∈ ran (projℎ‘𝐻)) |
14 | 10, 13 | eqeltrrd 2864 | . . 3 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ran (projℎ‘𝐻)) |
15 | 14 | ssriv 3861 | . 2 ⊢ 𝐻 ⊆ ran (projℎ‘𝐻) |
16 | 8, 15 | eqssi 3873 | 1 ⊢ ran (projℎ‘𝐻) = 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2048 ∀wral 3085 ⊆ wss 3828 ran crn 5405 Fn wfn 6181 ⟶wf 6182 ‘cfv 6186 ℋchba 28469 Cℋ cch 28479 projℎcpjh 28487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-rep 5047 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 ax-inf2 8894 ax-cc 9651 ax-cnex 10387 ax-resscn 10388 ax-1cn 10389 ax-icn 10390 ax-addcl 10391 ax-addrcl 10392 ax-mulcl 10393 ax-mulrcl 10394 ax-mulcom 10395 ax-addass 10396 ax-mulass 10397 ax-distr 10398 ax-i2m1 10399 ax-1ne0 10400 ax-1rid 10401 ax-rnegex 10402 ax-rrecex 10403 ax-cnre 10404 ax-pre-lttri 10405 ax-pre-lttrn 10406 ax-pre-ltadd 10407 ax-pre-mulgt0 10408 ax-pre-sup 10409 ax-addf 10410 ax-mulf 10411 ax-hilex 28549 ax-hfvadd 28550 ax-hvcom 28551 ax-hvass 28552 ax-hv0cl 28553 ax-hvaddid 28554 ax-hfvmul 28555 ax-hvmulid 28556 ax-hvmulass 28557 ax-hvdistr1 28558 ax-hvdistr2 28559 ax-hvmul0 28560 ax-hfi 28629 ax-his1 28632 ax-his2 28633 ax-his3 28634 ax-his4 28635 ax-hcompl 28752 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-nel 3071 df-ral 3090 df-rex 3091 df-reu 3092 df-rmo 3093 df-rab 3094 df-v 3414 df-sbc 3681 df-csb 3786 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-pss 3844 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-tp 4444 df-op 4446 df-uni 4711 df-int 4748 df-iun 4792 df-iin 4793 df-br 4928 df-opab 4990 df-mpt 5007 df-tr 5029 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7498 df-2nd 7499 df-supp 7631 df-wrecs 7747 df-recs 7809 df-rdg 7847 df-1o 7901 df-2o 7902 df-oadd 7905 df-omul 7906 df-er 8085 df-map 8204 df-pm 8205 df-ixp 8256 df-en 8303 df-dom 8304 df-sdom 8305 df-fin 8306 df-fsupp 8625 df-fi 8666 df-sup 8697 df-inf 8698 df-oi 8765 df-card 9158 df-acn 9161 df-cda 9384 df-pnf 10472 df-mnf 10473 df-xr 10474 df-ltxr 10475 df-le 10476 df-sub 10668 df-neg 10669 df-div 11095 df-nn 11436 df-2 11500 df-3 11501 df-4 11502 df-5 11503 df-6 11504 df-7 11505 df-8 11506 df-9 11507 df-n0 11705 df-z 11791 df-dec 11909 df-uz 12056 df-q 12160 df-rp 12202 df-xneg 12321 df-xadd 12322 df-xmul 12323 df-ioo 12555 df-ico 12557 df-icc 12558 df-fz 12706 df-fzo 12847 df-fl 12974 df-seq 13182 df-exp 13242 df-hash 13503 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-clim 14700 df-rlim 14701 df-sum 14898 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-cn 21533 df-cnp 21534 df-lm 21535 df-haus 21621 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-xms 22627 df-ms 22628 df-tms 22629 df-cfil 23555 df-cau 23556 df-cmet 23557 df-grpo 28041 df-gid 28042 df-ginv 28043 df-gdiv 28044 df-ablo 28093 df-vc 28107 df-nv 28140 df-va 28143 df-ba 28144 df-sm 28145 df-0v 28146 df-vs 28147 df-nmcv 28148 df-ims 28149 df-dip 28249 df-ssp 28270 df-ph 28361 df-cbn 28412 df-hnorm 28518 df-hba 28519 df-hvsub 28521 df-hlim 28522 df-hcau 28523 df-sh 28757 df-ch 28771 df-oc 28802 df-ch0 28803 df-shs 28860 df-pjh 28947 |
This theorem is referenced by: pjfoi 29255 pjfi 29256 pj11i 29263 pjss1coi 29715 pjimai 29728 pj3i 29760 |
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