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Mirrors > Home > MPE Home > Th. List > coeq0 | Structured version Visualization version GIF version |
Description: A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 6269 and coundir 6270 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
Ref | Expression |
---|---|
coeq0 | ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 6129 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relrn0 5986 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅) |
4 | rnco 6274 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
5 | 4 | eqeq1i 2740 | . 2 ⊢ (ran (𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅) |
6 | relres 6026 | . . . 4 ⊢ Rel (𝐴 ↾ ran 𝐵) | |
7 | reldm0 5941 | . . . 4 ⊢ (Rel (𝐴 ↾ ran 𝐵) → ((𝐴 ↾ ran 𝐵) = ∅ ↔ dom (𝐴 ↾ ran 𝐵) = ∅)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ ((𝐴 ↾ ran 𝐵) = ∅ ↔ dom (𝐴 ↾ ran 𝐵) = ∅) |
9 | relrn0 5986 | . . . 4 ⊢ (Rel (𝐴 ↾ ran 𝐵) → ((𝐴 ↾ ran 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)) | |
10 | 6, 9 | ax-mp 5 | . . 3 ⊢ ((𝐴 ↾ ran 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅) |
11 | dmres 6032 | . . . . 5 ⊢ dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
12 | incom 4217 | . . . . 5 ⊢ (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵) | |
13 | 11, 12 | eqtri 2763 | . . . 4 ⊢ dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵) |
14 | 13 | eqeq1i 2740 | . . 3 ⊢ (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) |
15 | 8, 10, 14 | 3bitr3i 301 | . 2 ⊢ (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) |
16 | 3, 5, 15 | 3bitri 297 | 1 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∩ cin 3962 ∅c0 4339 dom cdm 5689 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 |
This theorem is referenced by: coemptyd 15015 wrdpmtrlast 33096 diophrw 42747 relexpnul 43668 |
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