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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 2727 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
3 | eqid 2727 | . . . 4 β’ (ocvβπ) = (ocvβπ) | |
4 | eqid 2727 | . . . 4 β’ (proj1βπ) = (proj1βπ) | |
5 | pjf.k | . . . 4 β’ πΎ = (projβπ) | |
6 | 1, 2, 3, 4, 5 | pjdm 21634 | . . 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 21637 | . . 3 β’ (π β dom πΎ β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
9 | 8 | feq1d 6701 | . 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 1534 β wcel 2099 dom cdm 5672 βΆwf 6538 βcfv 6542 (class class class)co 7414 Basecbs 17173 proj1cpj1 19583 LSubSpclss 20808 ocvcocv 21585 projcpj 21627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-pj 21630 |
This theorem is referenced by: (None) |
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