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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 6221 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
2 | uniss 4874 | . . . 4 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → ∪ 𝑅 ⊆ ∪ (dom 𝑅 × ran 𝑅)) | |
3 | uniss 4874 | . . . 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 5767 | . . 3 ⊢ ∪ ∪ (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅) | |
6 | 4, 5 | sstrdi 3957 | . 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)) |
7 | dmrnssfld 5926 | . . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
8 | 7 | a1i 11 | . 2 ⊢ (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅) |
9 | 6, 8 | eqssd 3962 | 1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∪ cun 3909 ⊆ wss 3911 ∪ cuni 4866 × cxp 5632 dom cdm 5634 ran crn 5635 Rel wrel 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: relresfld 6229 unidmrn 6232 relcnvfld 6233 unixp 6235 relexp0 14914 relexpfld 14940 rtrclreclem4 14952 dfrtrcl2 14953 lefld 18486 fvmptiunrelexplb0da 42045 |
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