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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 6793 | . . . 4 ⊢ (𝐹:𝐷–1-1→𝑆 → 𝐹:𝐷⟶𝑆) | |
2 | fcoi2 6772 | . . . 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 1132 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → (( I ↾ 𝑆) ∘ 𝐹) = 𝐹) |
5 | simp1 1133 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝑊 ∈ LMod) | |
6 | linds2 21762 | . . . 4 ⊢ (𝑆 ∈ (LIndS‘𝑊) → ( I ↾ 𝑆) LIndF 𝑊) | |
7 | 6 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → ( I ↾ 𝑆) LIndF 𝑊) |
8 | dmresi 6056 | . . . . . 6 ⊢ dom ( I ↾ 𝑆) = 𝑆 | |
9 | f1eq3 6790 | . . . . . 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 227 | . . . 4 ⊢ (𝐹:𝐷–1-1→𝑆 → 𝐹:𝐷–1-1→dom ( I ↾ 𝑆)) |
12 | 11 | 3ad2ant3 1132 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹:𝐷–1-1→dom ( I ↾ 𝑆)) |
13 | f1lindf 21773 | . . 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 1368 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → (( I ↾ 𝑆) ∘ 𝐹) LIndF 𝑊) |
15 | 4, 14 | eqbrtrrd 5173 | 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 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 I cid 5575 dom cdm 5678 ↾ cres 5680 ∘ ccom 5682 ⟶wf 6545 –1-1→wf1 6546 ‘cfv 6549 LModclmod 20755 LIndF clindf 21755 LIndSclinds 21756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-1cn 11198 ax-addcl 11200 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12246 df-slot 17154 df-ndx 17166 df-base 17184 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lindf 21757 df-linds 21758 |
This theorem is referenced by: islindf3 21777 lindsmm 21779 lbslcic 21792 |
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