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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 6398 | . . . 4 ⊢ (𝐹:𝐷–1-1→𝑆 → 𝐹:𝐷⟶𝑆) | |
2 | fcoi2 6376 | . . . 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 1115 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → (( I ↾ 𝑆) ∘ 𝐹) = 𝐹) |
5 | simp1 1116 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝑊 ∈ LMod) | |
6 | linds2 20647 | . . . 4 ⊢ (𝑆 ∈ (LIndS‘𝑊) → ( I ↾ 𝑆) LIndF 𝑊) | |
7 | 6 | 3ad2ant2 1114 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → ( I ↾ 𝑆) LIndF 𝑊) |
8 | dmresi 5757 | . . . . . 6 ⊢ dom ( I ↾ 𝑆) = 𝑆 | |
9 | f1eq3 6395 | . . . . . 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 220 | . . . 4 ⊢ (𝐹:𝐷–1-1→𝑆 → 𝐹:𝐷–1-1→dom ( I ↾ 𝑆)) |
12 | 11 | 3ad2ant3 1115 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹:𝐷–1-1→dom ( I ↾ 𝑆)) |
13 | f1lindf 20658 | . . 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 1351 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → (( I ↾ 𝑆) ∘ 𝐹) LIndF 𝑊) |
15 | 4, 14 | eqbrtrrd 4947 | 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 198 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 class class class wbr 4923 I cid 5304 dom cdm 5400 ↾ cres 5402 ∘ ccom 5404 ⟶wf 6178 –1-1→wf1 6179 ‘cfv 6182 LModclmod 19346 LIndF clindf 20640 LIndSclinds 20641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-slot 16333 df-base 16335 df-0g 16561 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-lmod 19348 df-lss 19416 df-lsp 19456 df-lindf 20642 df-linds 20643 |
This theorem is referenced by: islindf3 20662 lindsmm 20664 lbslcic 20677 |
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