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Mirrors > Home > MPE Home > Th. List > quslem | Structured version Visualization version GIF version |
Description: The function in qusval 17488 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
qusval.u | β’ (π β π = (π /s βΌ )) |
qusval.v | β’ (π β π = (Baseβπ )) |
qusval.f | β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) |
qusval.e | β’ (π β βΌ β π) |
qusval.r | β’ (π β π β π) |
Ref | Expression |
---|---|
quslem | β’ (π β πΉ:πβontoβ(π / βΌ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusval.e | . . . . . 6 β’ (π β βΌ β π) | |
2 | ecexg 8707 | . . . . . 6 β’ ( βΌ β π β [π₯] βΌ β V) | |
3 | 1, 2 | syl 17 | . . . . 5 β’ (π β [π₯] βΌ β V) |
4 | 3 | ralrimivw 3151 | . . . 4 β’ (π β βπ₯ β π [π₯] βΌ β V) |
5 | qusval.f | . . . . 5 β’ πΉ = (π₯ β π β¦ [π₯] βΌ ) | |
6 | 5 | fnmpt 6691 | . . . 4 β’ (βπ₯ β π [π₯] βΌ β V β πΉ Fn π) |
7 | 4, 6 | syl 17 | . . 3 β’ (π β πΉ Fn π) |
8 | dffn4 6812 | . . 3 β’ (πΉ Fn π β πΉ:πβontoβran πΉ) | |
9 | 7, 8 | sylib 217 | . 2 β’ (π β πΉ:πβontoβran πΉ) |
10 | 5 | rnmpt 5955 | . . . 4 β’ ran πΉ = {π¦ β£ βπ₯ β π π¦ = [π₯] βΌ } |
11 | df-qs 8709 | . . . 4 β’ (π / βΌ ) = {π¦ β£ βπ₯ β π π¦ = [π₯] βΌ } | |
12 | 10, 11 | eqtr4i 2764 | . . 3 β’ ran πΉ = (π / βΌ ) |
13 | foeq3 6804 | . . 3 β’ (ran πΉ = (π / βΌ ) β (πΉ:πβontoβran πΉ β πΉ:πβontoβ(π / βΌ ))) | |
14 | 12, 13 | ax-mp 5 | . 2 β’ (πΉ:πβontoβran πΉ β πΉ:πβontoβ(π / βΌ )) |
15 | 9, 14 | sylib 217 | 1 β’ (π β πΉ:πβontoβ(π / βΌ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 {cab 2710 βwral 3062 βwrex 3071 Vcvv 3475 β¦ cmpt 5232 ran crn 5678 Fn wfn 6539 βontoβwfo 6542 βcfv 6544 (class class class)co 7409 [cec 8701 / cqs 8702 Basecbs 17144 /s cqus 17451 |
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 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-fun 6546 df-fn 6547 df-fo 6550 df-ec 8705 df-qs 8709 |
This theorem is referenced by: qusbas 17491 quss 17492 qusaddvallem 17497 qusaddflem 17498 qusaddval 17499 qusaddf 17500 qusmulval 17501 qusmulf 17502 qusgrp2 18941 qusring2 20147 znzrhfo 21103 qustps 23226 qustgpopn 23624 qustgplem 23625 qustgphaus 23627 qusker 32464 qusvsval 32467 quslmod 32469 quslmhm 32470 qusdimsum 32713 qusrng 46681 |
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