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Mirrors > Home > HSE Home > Th. List > pjfi | Structured version Visualization version GIF version |
Description: The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjfn.1 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjfi | ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjfn.1 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
2 | 1 | pjfni 31389 | . 2 ⊢ (projℎ‘𝐻) Fn ℋ |
3 | 1 | pjrni 31390 | . . 3 ⊢ ran (projℎ‘𝐻) = 𝐻 |
4 | 1 | chssii 30919 | . . 3 ⊢ 𝐻 ⊆ ℋ |
5 | 3, 4 | eqsstri 4008 | . 2 ⊢ ran (projℎ‘𝐻) ⊆ ℋ |
6 | df-f 6537 | . 2 ⊢ ((projℎ‘𝐻): ℋ⟶ ℋ ↔ ((projℎ‘𝐻) Fn ℋ ∧ ran (projℎ‘𝐻) ⊆ ℋ)) | |
7 | 2, 5, 6 | mpbir2an 708 | 1 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ⊆ wss 3940 ran crn 5667 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 ℋchba 30607 Cℋ cch 30617 projℎcpjh 30625 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cc 10425 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 ax-hilex 30687 ax-hfvadd 30688 ax-hvcom 30689 ax-hvass 30690 ax-hv0cl 30691 ax-hvaddid 30692 ax-hfvmul 30693 ax-hvmulid 30694 ax-hvmulass 30695 ax-hvdistr1 30696 ax-hvdistr2 30697 ax-hvmul0 30698 ax-hfi 30767 ax-his1 30770 ax-his2 30771 ax-his3 30772 ax-his4 30773 ax-hcompl 30890 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-acn 9932 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-cn 23052 df-cnp 23053 df-lm 23054 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cfil 25104 df-cau 25105 df-cmet 25106 df-grpo 30181 df-gid 30182 df-ginv 30183 df-gdiv 30184 df-ablo 30233 df-vc 30247 df-nv 30280 df-va 30283 df-ba 30284 df-sm 30285 df-0v 30286 df-vs 30287 df-nmcv 30288 df-ims 30289 df-dip 30389 df-ssp 30410 df-ph 30501 df-cbn 30551 df-hnorm 30656 df-hba 30657 df-hvsub 30659 df-hlim 30660 df-hcau 30661 df-sh 30895 df-ch 30909 df-oc 30940 df-ch0 30941 df-shs 30996 df-pjh 31083 |
This theorem is referenced by: ho0f 31439 hoid1i 31477 hoid1ri 31478 pjhmopi 31834 pjnmopi 31836 pjsdii 31843 pjddii 31844 pjcoi 31846 pjcohcli 31848 pjcofni 31850 pjss1coi 31851 pjss2coi 31852 pjorthcoi 31857 pjscji 31858 pjssposi 31860 pjssdif2i 31862 pjtoi 31867 pjoci 31868 pjclem1 31883 pjclem3 31885 pjclem4 31887 pjci 31888 pjcohocli 31891 pjadj2coi 31892 pj2cocli 31893 pj3lem1 31894 pj3si 31895 pj3cor1i 31897 |
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