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Mirrors > Home > MPE Home > Th. List > relfld | Structured version Visualization version GIF version |
Description: The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
Ref | Expression |
---|---|
relfld | ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relssdmrn 6088 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
2 | uniss 4808 | . . . 4 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → ∪ 𝑅 ⊆ ∪ (dom 𝑅 × ran 𝑅)) | |
3 | uniss 4808 | . . . 4 ⊢ (∪ 𝑅 ⊆ ∪ (dom 𝑅 × ran 𝑅) → ∪ ∪ 𝑅 ⊆ ∪ ∪ (dom 𝑅 × ran 𝑅)) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 ⊆ ∪ ∪ (dom 𝑅 × ran 𝑅)) |
5 | unixpss 5647 | . . 3 ⊢ ∪ ∪ (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅) | |
6 | 4, 5 | sstrdi 3927 | . 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)) |
7 | dmrnssfld 5806 | . . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
8 | 7 | a1i 11 | . 2 ⊢ (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅) |
9 | 6, 8 | eqssd 3932 | 1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∪ cun 3879 ⊆ wss 3881 ∪ cuni 4800 × cxp 5517 dom cdm 5519 ran crn 5520 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-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: relresfld 6095 unidmrn 6098 relcnvfld 6099 unixp 6101 relexp0 14374 relexpfld 14400 rtrclreclem4 14412 dfrtrcl2 14413 lefld 17828 fvmptiunrelexplb0da 40386 |
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