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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 40029 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:𝐵–1-1-onto→𝐵) |
4 | f1of1 6842 | . . 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 7276 | . 2 ⊢ ((𝐹:𝐵–1-1→𝐵 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ 𝑋 = 𝑌)) | |
7 | 5, 6 | sylan 579 | 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 1535 ∈ wcel 2104 –1-1→wf1 6555 –1-1-onto→wf1o 6557 ‘cfv 6558 Basecbs 17234 LAutclaut 39929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 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 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-map 8861 df-laut 39933 |
This theorem is referenced by: lautlt 40035 ltrn11 40070 |
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