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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 2724 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
3 | eqid 2724 | . . . 4 β’ (ocvβπ) = (ocvβπ) | |
4 | eqid 2724 | . . . 4 β’ (proj1βπ) = (proj1βπ) | |
5 | pjf.k | . . . 4 β’ πΎ = (projβπ) | |
6 | 1, 2, 3, 4, 5 | pjdm 21570 | . . 3 β’ (π β dom πΎ β (π β (LSubSpβπ) β§ (π(proj1βπ)((ocvβπ)βπ)):πβΆπ)) |
7 | 6 | simprbi 496 | . 2 β’ (π β dom πΎ β (π(proj1βπ)((ocvβπ)βπ)):πβΆπ) |
8 | 3, 4, 5 | pjval 21573 | . . 3 β’ (π β dom πΎ β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
9 | 8 | feq1d 6692 | . 2 β’ (π β dom πΎ β ((πΎβπ):πβΆπ β (π(proj1βπ)((ocvβπ)βπ)):πβΆπ)) |
10 | 7, 9 | mpbird 257 | 1 β’ (π β dom πΎ β (πΎβπ):πβΆπ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 dom cdm 5666 βΆwf 6529 βcfv 6533 (class class class)co 7401 Basecbs 17143 proj1cpj1 19545 LSubSpclss 20768 ocvcocv 21521 projcpj 21563 |
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-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 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-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-map 8818 df-pj 21566 |
This theorem is referenced by: (None) |
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