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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 6237 and coundir 6238 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 6098 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relrn0 5959 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅) |
4 | rnco 6242 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
5 | 4 | eqeq1i 2729 | . 2 ⊢ (ran (𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅) |
6 | relres 6001 | . . . 4 ⊢ Rel (𝐴 ↾ ran 𝐵) | |
7 | reldm0 5918 | . . . 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 5959 | . . . 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 5994 | . . . . 5 ⊢ dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
12 | incom 4194 | . . . . 5 ⊢ (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵) | |
13 | 11, 12 | eqtri 2752 | . . . 4 ⊢ dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵) |
14 | 13 | eqeq1i 2729 | . . 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 205 = wceq 1533 ∩ cin 3940 ∅c0 4315 dom cdm 5667 ran crn 5668 ↾ cres 5669 ∘ ccom 5671 Rel wrel 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 |
This theorem is referenced by: coemptyd 14928 diophrw 42047 relexpnul 42978 |
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