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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 5901 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
2 | uniss 4683 | . . . 4 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → ∪ 𝑅 ⊆ ∪ (dom 𝑅 × ran 𝑅)) | |
3 | uniss 4683 | . . . 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 5472 | . . 3 ⊢ ∪ ∪ (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅) | |
6 | 4, 5 | syl6ss 3839 | . 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)) |
7 | dmrnssfld 5621 | . . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
8 | 7 | a1i 11 | . 2 ⊢ (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅) |
9 | 6, 8 | eqssd 3844 | 1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∪ cun 3796 ⊆ wss 3798 ∪ cuni 4660 × cxp 5344 dom cdm 5346 ran crn 5347 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-dm 5356 df-rn 5357 |
This theorem is referenced by: relresfld 5907 unidmrn 5910 relcnvfld 5911 unixp 5913 relexp0 14147 relexpfld 14173 rtrclreclem4 14185 dfrtrcl2 14186 lefld 17586 fvmptiunrelexplb0da 38813 |
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