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Mirrors > Home > MPE Home > Th. List > relcnvfld | Structured version Visualization version GIF version |
Description: if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Ref | Expression |
---|---|
relcnvfld | ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfld 6177 | . 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | |
2 | unidmrn 6181 | . 2 ⊢ ∪ ∪ ◡𝑅 = (dom 𝑅 ∪ ran 𝑅) | |
3 | 1, 2 | eqtr4di 2798 | 1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∪ cun 3890 ∪ cuni 4845 ◡ccnv 5589 dom cdm 5590 ran crn 5591 Rel wrel 5595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-xp 5596 df-rel 5597 df-cnv 5598 df-dm 5600 df-rn 5601 |
This theorem is referenced by: cnvps 18294 tsrdir 18320 |
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