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| 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 6288 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
| 2 | uniss 4915 | . . . 4 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → ∪ 𝑅 ⊆ ∪ (dom 𝑅 × ran 𝑅)) | |
| 3 | uniss 4915 | . . . 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 5820 | . . 3 ⊢ ∪ ∪ (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅) | |
| 6 | 4, 5 | sstrdi 3996 | . 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)) | 
| 7 | dmrnssfld 5984 | . . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
| 8 | 7 | a1i 11 | . 2 ⊢ (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅) | 
| 9 | 6, 8 | eqssd 4001 | 1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∪ cun 3949 ⊆ wss 3951 ∪ cuni 4907 × cxp 5683 dom cdm 5685 ran crn 5686 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 | 
| This theorem is referenced by: relresfld 6296 unidmrn 6299 relcnvfld 6300 unixp 6302 relexp0 15062 relexpfld 15088 rtrclreclem4 15100 dfrtrcl2 15101 lefld 18637 fvmptiunrelexplb0da 43698 | 
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