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Mirrors > Home > MPE Home > Th. List > pjval | Structured version Visualization version GIF version |
Description: Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjfval2.o | β’ β₯ = (ocvβπ) |
pjfval2.p | β’ π = (proj1βπ) |
pjfval2.k | β’ πΎ = (projβπ) |
Ref | Expression |
---|---|
pjval | β’ (π β dom πΎ β (πΎβπ) = (ππ( β₯ βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 β’ (π₯ = π β π₯ = π) | |
2 | fveq2 6843 | . . 3 β’ (π₯ = π β ( β₯ βπ₯) = ( β₯ βπ)) | |
3 | 1, 2 | oveq12d 7376 | . 2 β’ (π₯ = π β (π₯π( β₯ βπ₯)) = (ππ( β₯ βπ))) |
4 | pjfval2.o | . . 3 β’ β₯ = (ocvβπ) | |
5 | pjfval2.p | . . 3 β’ π = (proj1βπ) | |
6 | pjfval2.k | . . 3 β’ πΎ = (projβπ) | |
7 | 4, 5, 6 | pjfval2 21131 | . 2 β’ πΎ = (π₯ β dom πΎ β¦ (π₯π( β₯ βπ₯))) |
8 | ovex 7391 | . 2 β’ (ππ( β₯ βπ)) β V | |
9 | 3, 7, 8 | fvmpt 6949 | 1 β’ (π β dom πΎ β (πΎβπ) = (ππ( β₯ βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 dom cdm 5634 βcfv 6497 (class class class)co 7358 proj1cpj1 19422 ocvcocv 21080 projcpj 21122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8770 df-pj 21125 |
This theorem is referenced by: pjf 21135 pjf2 21136 pjfo 21137 |
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