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Mirrors > Home > MPE Home > Th. List > f1linds | Structured version Visualization version GIF version |
Description: A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
f1linds | β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β πΉ LIndF π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6739 | . . . 4 β’ (πΉ:π·β1-1βπ β πΉ:π·βΆπ) | |
2 | fcoi2 6718 | . . . 4 β’ (πΉ:π·βΆπ β (( I βΎ π) β πΉ) = πΉ) | |
3 | 1, 2 | syl 17 | . . 3 β’ (πΉ:π·β1-1βπ β (( I βΎ π) β πΉ) = πΉ) |
4 | 3 | 3ad2ant3 1136 | . 2 β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β (( I βΎ π) β πΉ) = πΉ) |
5 | simp1 1137 | . . 3 β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β π β LMod) | |
6 | linds2 21233 | . . . 4 β’ (π β (LIndSβπ) β ( I βΎ π) LIndF π) | |
7 | 6 | 3ad2ant2 1135 | . . 3 β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β ( I βΎ π) LIndF π) |
8 | dmresi 6006 | . . . . . 6 β’ dom ( I βΎ π) = π | |
9 | f1eq3 6736 | . . . . . 6 β’ (dom ( I βΎ π) = π β (πΉ:π·β1-1βdom ( I βΎ π) β πΉ:π·β1-1βπ)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 β’ (πΉ:π·β1-1βdom ( I βΎ π) β πΉ:π·β1-1βπ) |
11 | 10 | biimpri 227 | . . . 4 β’ (πΉ:π·β1-1βπ β πΉ:π·β1-1βdom ( I βΎ π)) |
12 | 11 | 3ad2ant3 1136 | . . 3 β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β πΉ:π·β1-1βdom ( I βΎ π)) |
13 | f1lindf 21244 | . . 3 β’ ((π β LMod β§ ( I βΎ π) LIndF π β§ πΉ:π·β1-1βdom ( I βΎ π)) β (( I βΎ π) β πΉ) LIndF π) | |
14 | 5, 7, 12, 13 | syl3anc 1372 | . 2 β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β (( I βΎ π) β πΉ) LIndF π) |
15 | 4, 14 | eqbrtrrd 5130 | 1 β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β πΉ LIndF π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 I cid 5531 dom cdm 5634 βΎ cres 5636 β ccom 5638 βΆwf 6493 β1-1βwf1 6494 βcfv 6497 LModclmod 20336 LIndF clindf 21226 LIndSclinds 21227 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-1cn 11114 ax-addcl 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 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-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12159 df-slot 17059 df-ndx 17071 df-base 17089 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lindf 21228 df-linds 21229 |
This theorem is referenced by: islindf3 21248 lindsmm 21250 lbslcic 21263 |
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