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Mirrors > Home > MPE Home > Th. List > Mathboxes > laut11 | Structured version Visualization version GIF version |
Description: One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
laut1o.b | ⊢ 𝐵 = (Base‘𝐾) |
laut1o.i | ⊢ 𝐼 = (LAut‘𝐾) |
Ref | Expression |
---|---|
laut11 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | laut1o.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | laut1o.i | . . . 4 ⊢ 𝐼 = (LAut‘𝐾) | |
3 | 1, 2 | laut1o 40065 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:𝐵–1-1-onto→𝐵) |
4 | f1of1 6845 | . . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵–1-1→𝐵) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:𝐵–1-1→𝐵) |
6 | f1fveq 7280 | . 2 ⊢ ((𝐹:𝐵–1-1→𝐵 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ 𝑋 = 𝑌)) | |
7 | 5, 6 | sylan 580 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 –1-1→wf1 6556 –1-1-onto→wf1o 6558 ‘cfv 6559 Basecbs 17243 LAutclaut 39965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-map 8864 df-laut 39969 |
This theorem is referenced by: lautlt 40071 ltrn11 40106 |
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