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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 | ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6101 | . . . 4 ⊢ (𝐹:𝐷–1-1→𝑆 → 𝐹:𝐷⟶𝑆) | |
2 | fcoi2 6079 | . . . 4 ⊢ (𝐹:𝐷⟶𝑆 → (( I ↾ 𝑆) ∘ 𝐹) = 𝐹) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹:𝐷–1-1→𝑆 → (( I ↾ 𝑆) ∘ 𝐹) = 𝐹) |
4 | 3 | 3ad2ant3 1084 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → (( I ↾ 𝑆) ∘ 𝐹) = 𝐹) |
5 | simp1 1061 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝑊 ∈ LMod) | |
6 | linds2 20150 | . . . 4 ⊢ (𝑆 ∈ (LIndS‘𝑊) → ( I ↾ 𝑆) LIndF 𝑊) | |
7 | 6 | 3ad2ant2 1083 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → ( I ↾ 𝑆) LIndF 𝑊) |
8 | dmresi 5457 | . . . . . 6 ⊢ dom ( I ↾ 𝑆) = 𝑆 | |
9 | f1eq3 6098 | . . . . . 6 ⊢ (dom ( I ↾ 𝑆) = 𝑆 → (𝐹:𝐷–1-1→dom ( I ↾ 𝑆) ↔ 𝐹:𝐷–1-1→𝑆)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (𝐹:𝐷–1-1→dom ( I ↾ 𝑆) ↔ 𝐹:𝐷–1-1→𝑆) |
11 | 10 | biimpri 218 | . . . 4 ⊢ (𝐹:𝐷–1-1→𝑆 → 𝐹:𝐷–1-1→dom ( I ↾ 𝑆)) |
12 | 11 | 3ad2ant3 1084 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹:𝐷–1-1→dom ( I ↾ 𝑆)) |
13 | f1lindf 20161 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ ( I ↾ 𝑆) LIndF 𝑊 ∧ 𝐹:𝐷–1-1→dom ( I ↾ 𝑆)) → (( I ↾ 𝑆) ∘ 𝐹) LIndF 𝑊) | |
14 | 5, 7, 12, 13 | syl3anc 1326 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → (( I ↾ 𝑆) ∘ 𝐹) LIndF 𝑊) |
15 | 4, 14 | eqbrtrrd 4677 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 I cid 5023 dom cdm 5114 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 LModclmod 18863 LIndF clindf 20143 LIndSclinds 20144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-slot 15861 df-base 15863 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lindf 20145 df-linds 20146 |
This theorem is referenced by: islindf3 20165 lindsmm 20167 lbslcic 20180 |
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