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Mirrors > Home > MPE Home > Th. List > Mathboxes > pats | Structured version Visualization version GIF version |
Description: The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.) |
Ref | Expression |
---|---|
patoms.b | ⊢ 𝐵 = (Base‘𝐾) |
patoms.z | ⊢ 0 = (0.‘𝐾) |
patoms.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
patoms.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
pats | ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3514 | . 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V) | |
2 | patoms.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | fveq2 6672 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾)) | |
4 | patoms.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | syl6eqr 2876 | . . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵) |
6 | fveq2 6672 | . . . . . . . 8 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾)) | |
7 | patoms.c | . . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
8 | 6, 7 | syl6eqr 2876 | . . . . . . 7 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶) |
9 | 8 | breqd 5079 | . . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥)) |
10 | fveq2 6672 | . . . . . . . 8 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾)) | |
11 | patoms.z | . . . . . . . 8 ⊢ 0 = (0.‘𝐾) | |
12 | 10, 11 | syl6eqr 2876 | . . . . . . 7 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 ) |
13 | 12 | breq1d 5078 | . . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥 ↔ 0 𝐶𝑥)) |
14 | 9, 13 | bitrd 281 | . . . . 5 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ 0 𝐶𝑥)) |
15 | 5, 14 | rabeqbidv 3487 | . . . 4 ⊢ (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
16 | df-ats 36405 | . . . 4 ⊢ Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥}) | |
17 | 4 | fvexi 6686 | . . . . 5 ⊢ 𝐵 ∈ V |
18 | 17 | rabex 5237 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ∈ V |
19 | 15, 16, 18 | fvmpt 6770 | . . 3 ⊢ (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
20 | 2, 19 | syl5eq 2870 | . 2 ⊢ (𝐾 ∈ V → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
21 | 1, 20 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 0.cp0 17649 ⋖ ccvr 36400 Atomscatm 36401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ats 36405 |
This theorem is referenced by: isat 36424 |
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