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Mirrors > Home > MPE Home > Th. List > pjf | Structured version Visualization version GIF version |
Description: A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjf.k | β’ πΎ = (projβπ) |
pjf.v | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
pjf | β’ (π β dom πΎ β (πΎβπ):πβΆπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjf.v | . . . 4 β’ π = (Baseβπ) | |
2 | eqid 2732 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
3 | eqid 2732 | . . . 4 β’ (ocvβπ) = (ocvβπ) | |
4 | eqid 2732 | . . . 4 β’ (proj1βπ) = (proj1βπ) | |
5 | pjf.k | . . . 4 β’ πΎ = (projβπ) | |
6 | 1, 2, 3, 4, 5 | pjdm 21261 | . . 3 β’ (π β dom πΎ β (π β (LSubSpβπ) β§ (π(proj1βπ)((ocvβπ)βπ)):πβΆπ)) |
7 | 6 | simprbi 497 | . 2 β’ (π β dom πΎ β (π(proj1βπ)((ocvβπ)βπ)):πβΆπ) |
8 | 3, 4, 5 | pjval 21264 | . . 3 β’ (π β dom πΎ β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
9 | 8 | feq1d 6702 | . 2 β’ (π β dom πΎ β ((πΎβπ):πβΆπ β (π(proj1βπ)((ocvβπ)βπ)):πβΆπ)) |
10 | 7, 9 | mpbird 256 | 1 β’ (π β dom πΎ β (πΎβπ):πβΆπ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 dom cdm 5676 βΆwf 6539 βcfv 6543 (class class class)co 7408 Basecbs 17143 proj1cpj1 19502 LSubSpclss 20541 ocvcocv 21212 projcpj 21254 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-map 8821 df-pj 21257 |
This theorem is referenced by: (None) |
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