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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 6067 and coundir 6068 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 6064 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relrn0 5805 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅) |
4 | rnco 6072 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
5 | 4 | eqeq1i 2803 | . 2 ⊢ (ran (𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅) |
6 | relres 5847 | . . . 4 ⊢ Rel (𝐴 ↾ ran 𝐵) | |
7 | reldm0 5762 | . . . 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 5805 | . . . 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 5840 | . . . . 5 ⊢ dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
12 | incom 4128 | . . . . 5 ⊢ (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵) | |
13 | 11, 12 | eqtri 2821 | . . . 4 ⊢ dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵) |
14 | 13 | eqeq1i 2803 | . . 3 ⊢ (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) |
15 | 8, 10, 14 | 3bitr3i 304 | . 2 ⊢ (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) |
16 | 3, 5, 15 | 3bitri 300 | 1 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∩ cin 3880 ∅c0 4243 dom cdm 5519 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 |
This theorem is referenced by: coemptyd 14330 diophrw 39700 relexpnul 40379 |
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